# 7.6: Systems of Measurement (Part 1)

Skills to Develop

• Make unit conversions in the U.S. system
• Use mixed units of measurement in the U.S. system
• Make unit conversions in the metric system
• Use mixed units of measurement in the metric system
• Convert between the U.S. and the metric systems of measurement
• Convert between Fahrenheit and Celsius temperatures

be prepared!

Before you get started, take this readiness quiz.

1. Multiply: 4.29(1000). If you missed this problem, review Example 5.3.8.
2. Simplify: (dfrac{30}{54}). If you missed this problem, review Example 4.3.2.
3. Multiply: (dfrac{7}{15} cdot dfrac{25}{28}). If you missed this problem, review Example 4.3.9.

In this section we will see how to convert among different types of units, such as feet to miles or kilograms to pounds. The basic idea in all of the unit conversions will be to use a form of 1, the multiplicative identity, to change the units but not the value of a quantity.

## Make Unit Conversions in the U.S. System

There are two systems of measurement commonly used around the world. Most countries use the metric system. The United States uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.

The U.S. system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart and gallons. Both the U.S. system and the metric system measure time in seconds, minutes, or hours.

The equivalencies among the basic units of the U.S. system of measurement are listed in Table (PageIndex{1}). The table also shows, in parentheses, the common abbreviations for each measurement.

Table (PageIndex{1})
U.S. System Units
LengthVolume

1 foot (ft) = 12 inches (in)

1 yard (yd) = 3 feet (ft)

1 mile (mi) = 5280 feet (ft)

3 teaspoons (t) = 1 tablespoon (T)

16 Tablespoons (T) = 1 cup (C)

1 cup (C) = 8 fluid ounces (fl oz)

1 pint (pt) = 2 cups (C)

1 quart (qt) = 2 pints (pt)

1 gallon (gal) = 4 quarts (qt)

WeightTime

1 pound (lb) = 16 ounces (oz)

1 ton = 2000 pounds (lb)

1 minute (min) = 60 seconds (s)

1 hour (h) = 60 minutes (min)

1 day = 24 hours (h)

1 week (wk) = 7 days

1 year (yr) = 365 days

In many real-life applications, we need to convert between units of measurement. We will use the identity property of multiplication to do these conversions. We’ll restate the Identity Property of Multiplication here for easy reference.

For any real number a,

[a cdot 1 = a qquad 1 cdot a = a]

To use the identity property of multiplication, we write 1 in a form that will help us convert the units. For example, suppose we want to convert inches to feet. We know that 1 foot is equal to 12 inches, so we can write 1 as the fraction (dfrac{1; ft}{12; in}). When we multiply by this fraction, we do not change the value but just change the units. But (dfrac{12; in}{1; ft}) also equals 1. How do we decide whether to multiply by (dfrac{1; ft}{12; in}) or (dfrac{12; in}{1; ft})? We choose the fraction that will make the units we want to convert from divide out. For example, suppose we wanted to convert 60 inches to feet. If we choose the fraction that has inches in the denominator, we can eliminate the inches.

[60; cancel{in} cdot dfrac{1; ft}{12; cancel{in}} = 5; ft]

On the other hand, if we wanted to convert 5 feet to inches, we would choose the fraction that has feet in the denominator.

[5; cancel{ft} cdot dfrac{12; in}{1; cancel{ft}} = 60; in]

We treat the unit words like factors and ‘divide out’ common units like we do common factors.

HOW TO: MAKE UNIT CONVERSIONS

Step 1. Multiply the measurement to be converted by 1; write 1 as a fraction relating the units given and the units needed.

Step 2. Multiply.

Step 3. Simplify the fraction, performing the indicated operations and removing the common units.

Example (PageIndex{1}):

Mary Anne is 66 inches tall. What is her height in feet?

Solution

 Convert 66 inches into feet. Multiply the measurement to be converted by 1. 66 inches • 1 Write 1 as a fraction relating the units given and the units needed. $$66; inches cdot dfrac{1; foot}{12; inches}$$ Multiply. $$dfrac{66; inches; cdot 1; foot}{12; inches}$$ Simplify the fraction. $$dfrac{66; cancel{inches}; cdot 1; foot}{12; cancel{inches}} = dfrac{66; feet}{12}$$ 5.5 feet

Notice that the when we simplified the fraction, we first divided out the inches. Mary Anne is 5.5 feet tall.

Exercise (PageIndex{1}):

Lexie is 30 inches tall. Convert her height to feet.

2.5 feet

Exercise (PageIndex{2}):

Rene bought a hose that is 18 yards long. Convert the length to feet.

54 feet

When we use the Identity Property of Multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.

Example (PageIndex{2}):

Ndula, an elephant at the San Diego Safari Park, weighs almost 3.2 tons. Convert her weight to pounds.

Figure (PageIndex{1}) (credit: Guldo Da Rozze, Flickr)

Solution

We will convert 3.2 tons into pounds, using the equivalencies in Table (PageIndex{1}). We will use the Identity Property of Multiplication, writing 1 as the fraction (dfrac{2000; pounds}{1; ton})

 Multiply the measurement to be converted by 1. 3.2 tons • 1 Write 1 as a fraction relating tons and pounds. $$3.2; tons; cdot dfrac{2000; lbs}{1; tons}$$ Simplify. $$dfrac{3.2; cancel{tons}; cdot 2000; lbs}{1; cancel{tons}}$$ Multiply. 6400 lbs

.Ndula weighs almost 6,400 pounds.

Exercise (PageIndex{3}):

Arnold’s SUV weighs about 4.3 tons. Convert the weight to pounds.

8600 pounds

Exercise (PageIndex{4}):

A cruise ship weighs 51,000 tons. Convert the weight to pounds.

102,000,000 pounds

Sometimes to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.

Example (PageIndex{3}):

Juliet is going with her family to their summer home. She will be away for 9 weeks. Convert the time to minutes.

Solution

To convert weeks into minutes, we will convert weeks to days, days to hours, and then hours to minutes. To do this, we will multiply by conversion factors of 1.

 Write 1 as (dfrac{7; days}{1; week}, dfrac{24; hours}{1; day}, dfrac{60; minutes}{1; hour}). $$dfrac{9; wk}{1} cdot dfrac{7; days}{1; wk} cdot dfrac{24; hr}{1; day} cdot dfrac{60; min}{1; hr}$$ Cancel common units. $$dfrac{9; cancel{wk}}{1} cdot dfrac{7; cancel{ extcolor{blue}{days}}}{1; cancel{wk}} cdot dfrac{24; cancel{ extcolor{red}{hr}}}{1; cancel{ extcolor{blue}{day}}} cdot dfrac{60; min}{1; cancel{ extcolor{red}{hr}}}$$ Multiply. $$dfrac{9 cdot 7 cdot 24 cdot cdot 60; min}{1 cdot 1 cdot 1 cdot 1} = 90,720; min$$

Juliet will be away for 90,720 minutes.

Exercise (PageIndex{5})

The distance between Earth and the moon is about 250,000 miles. Convert this length to yards.

440,000,000 yards

Exercise (PageIndex{6}):

A team of astronauts spends 15 weeks in space. Convert the time to minutes.

151,200 minutes

Example (PageIndex{4}):

How many fluid ounces are in 1 gallon of milk?

Figure (PageIndex{2}) (credit: www.bluewaikiki.com, Flickr)

Solution

Use conversion factors to get the right units: convert gallons to quarts, quarts to pints, pints to cups, and cups to fluid ounces.

 Multiply the measurement to be converted by 1. $$dfrac{1; gal}{1} cdot dfrac{4; qt}{1; gal} cdot dfrac{2; pt}{1; qt} cdot dfrac{2; C}{1; pt} cdot dfrac{8; fl; oz}{1; C}$$ Simplify. $$dfrac{1; cancel{gal}}{1} cdot dfrac{4; cancel{qt}}{1; cancel{gal}} cdot dfrac{2; cancel{pt}}{1; cancel{qt}} cdot dfrac{2; cancel{C}}{1; cancel{pt}} cdot dfrac{8; fl; oz}{1; cancel{C}}$$ Multiply. $$dfrac{1 cdot 4 cdot 2 cdot 2 cdot 8; fl; oz}{1 cdot 1 cdot 1 cdot 1 cdot 1}$$ Simplify. 128 fluid ounces

There are 128 fluid ounces in a gallon.

Exercise (PageIndex{7}):

How many cups are in 1 gallon?

16 cups

Exercise (PageIndex{8}):

How many teaspoons are in 1 cup?

48 teaspoons

## Use Mixed Units of Measurement in the U.S. System

Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units.

Example (PageIndex{5}):

Charlie bought three steaks for a barbecue. Their weights were 14 ounces, 1 pound 2 ounces, and 1 pound 6 ounces. How many total pounds of steak did he buy?

Figure (PageIndex{3}) (credit: Helen Penjam, Flickr)

Solution

We will add the weights of the steaks to find the total weight of the steaks.

 Add the ounces. Then add the pounds. $$egin{split} 14; &ounces 1; pound quad 2; &ounces +; 1; pound quad 6; &ounces hline 2; pounds quad 22; &ounces end{split}$$ Convert 22 ounces to pounds and ounces. $$22; ounces = 1; pound,; 6; ounces$$ Add the pounds. 2 pounds + 1 pound, 6 ounces = 3 pounds, 6 ounces

Charlie bought 3 pounds 6 ounces of steak.

Exercise (PageIndex{9}):

Laura gave birth to triplets weighing 3 pounds 12 ounces, 3 pounds 3 ounces, and 2 pounds 9 ounces. What was the total birth weight of the three babies?

9 lbs. 8 oz

Exercise (PageIndex{10}):

Seymour cut two pieces of crown molding for his family room that were 8 feet 7 inches and 12 feet 11 inches. What was the total length of the molding?

21 ft. 6 in.

Example (PageIndex{6}):

Anthony bought four planks of wood that were each 6 feet 4 inches long. If the four planks are placed end-to-end, what is the total length of the wood?

Solution

We will multiply the length of one plank by 4 to find the total length.

 Multiply the inches and then the feet. $$egin{split} 6; feet quad 4; inches& imes qquad 4& hline 24; feet quad 16; inches& end{split}$$ Convert 16 inches to feet. 24 feet + 1 foot 4 inches Add the feet. 25 feet 4 inches

Anthony bought 25 feet 4 inches of wood.

Exercise (PageIndex{11}):

Henri wants to triple his spaghetti sauce recipe, which calls for 1 pound 8 ounces of ground turkey. How many pounds of ground turkey will he need?

4 lbs. 8 oz.

Exercise (PageIndex{12}):

Joellen wants to double a solution of 5 gallons 3 quarts. How many gallons of solution will she have in all?

11 gal. 2 qts.

## Make Unit Conversions in the Metric System

In the metric system, units are related by powers of 10. The root words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is 1000 meters; the prefix kilo- means thousand. One centimeter is (dfrac{1}{100}) of a meter, because the prefix centi- means one one-hundredth (just like one cent is (dfrac{1}{100}) of one dollar).

The equivalencies of measurements in the metric system are shown in Table (PageIndex{2}). The common abbreviations for each measurement are given in parentheses.

Table (PageIndex{2})
Metric Measurements
LengthMassVolume/Capacity

1 kilometer (km) = 1000 m

1 hectometer (hm) = 100 m

1 dekameter (dam) = 10 m

1 meter (m) = 1 m

1 decimeter (dm) = 0.1 m

1 centimeter (cm) = 0.01 m

1 millimeter (mm) = 0.001 m

1 kilogram (kg) = 1000 g

1 hectogram (hg) = 100 g

1 dekagram (dag) = 10 g

1 gram (g) = 1 g

1 decigram (dg) = 0.1 g

1 centigram (cg) = 0.01 g

1 milligram (mg) = 0.001 g

1 kiloliter (kL) = 1000 L

1 hectoliter (hL) = 100 L

1 dekaliter (daL) = 10 L

1 liter (L) = 1 L

1 deciliter (dL) = 0.1 L

1 centiliter (cL) = 0.01 L

1 milliliter (mL) = 0.001 L

1 meter = 100 centimeters

1 meter = 1000 millimeters

1 gram = 100 centigrams

1 gram = 1000 milligrams

1 liter = 100 centiliters

1 liter = 1000 milliliters

To make conversions in the metric system, we will use the same technique we did in the U.S. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.

Have you ever run a 5 k or 10 k race? The lengths of those races are measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.

Example (PageIndex{7}):

Nick ran a 10-kilometer race. How many meters did he run?

Figure (PageIndex{4}) (credit: William Warby, Flickr)

Solution

We will convert kilometers to meters using the Identity Property of Multiplication and the equivalencies in Table 7.63.

 Multiply the measurement to be converted by 1. $$10; extcolor{red}{km}; cdot 1$$ Write 1 as a fraction relating kilometers and meters. $$10; extcolor{red}{km}; cdot dfrac{1000; m}{1; extcolor{red}{km}}$$ Simplify. $$dfrac{10; cancel{ extcolor{red}{km}}; cdot 1000; m}{1; cancel{ extcolor{red}{km}}}$$ Multiply. $$10,000; m$$

Nick ran 10,000 meters.

Exercise (PageIndex{13}):

Sandy completed her first 5-km race. How many meters did she run?

5000 m

Exercise (PageIndex{14}):

Herman bought a rug 2.5 meters in length. How many centimeters is the length?

250 cm

Example (PageIndex{8}):

Eleanor’s newborn baby weighed 3200 grams. How many kilograms did the baby weigh?

Solution

We will convert grams to kilograms.

 Multiply the measurement to be converted by 1. $$3200; extcolor{red}{g}; cdot 1$$ Write 1 as a fraction relating kilograms and grams. $$3200; extcolor{red}{g}; cdot dfrac{1; kg}{1000; extcolor{red}{g}}$$ Simplify. $$3200; cancel{ extcolor{red}{g}}; cdot dfrac{1; kg}{1000; cancel{ extcolor{red}{g}}}$$ Multiply. $$dfrac{3200; kilograms}{1000}$$ Divide. $$3.2; kilograms$$

The baby weighed 3.2 kilograms.

Exercise (PageIndex{15})

Kari’s newborn baby weighed 2800 grams. How many kilograms did the baby weigh?

2.8 kilograms

Exercise (PageIndex{16})

Anderson received a package that was marked 45004500 grams. How many kilograms did this package weigh?

4.5 kilograms

Since the metric system is based on multiples of ten, conversions involve multiplying by multiples of ten. In Decimal Operations, we learned how to simplify these calculations by just moving the decimal. To multiply by 10, 100, or 1000, we move the decimal to the right 1, 2, or 3 places, respectively. To multiply by 0.1, 0.01, or 0.001 we move the decimal to the left 1, 2, or 3 places respectively. We can apply this pattern when we make measurement conversions in the metric system.

In Example 7.51, we changed 3200 grams to kilograms by multiplying by 1 1000 (or 0.001). This is the same as moving the decimal 3 places to the left.

Example (PageIndex{9}):

Convert: (a) 350 liters to kiloliters (b) 4.1 liters to milliliters.

Solution

(a) We will convert liters to kiloliters. In Table 7.63, we see that 1 kiloliter = 1000 liters.

 Multiply by 1, writing 1 as a fraction relating liters to kiloliters. $$350; L; cdot dfrac{1; kL}{1000; L}$$ Simplify. $$350; cancel{L}; cdot dfrac{1; kL}{1000; cancel{L}}$$ Move the decimal 3 units to the left.

0.35 kL

(b) We will convert liters to milliliters. In Table 7.63, we see that 1 liter = 1000 milliliters.

 Multiply by 1, writing 1 as a fraction relating milliliters to liters. $$4.1; L; cdot dfrac{1000; mL}{1; L}$$ Simplify. $$4.1; cancel{L}; cdot dfrac{1000; mL}{1; cancel{L}}$$ Move the decimal 3 units to the left.

4100 mL

Exercise (PageIndex{17}):

Convert: (a) 7.25 L to kL (b) 6.3 L to mL.

0.00725 kL

6300 mL

Exercise (PageIndex{18}):

Convert: (a) 350 hL to L (b) 4.1 L to cL.

35,000 L

410 cL

## Measurement System Analysis (MSA)

If measurements are used to guide decisions, then it follows logically that the more error there is in the measurements, the more error there will be in the decisions based on those measurements. The purpose of Measurement System Analysis is to qualify a measurement system for use by quantifying its accuracy, precision, and stability.

An example from industry serves to illustrate the importance of measurement system quality:

A manufacturer of building products was struggling to improve process yields, which had a significant impact on product cost. Experience indicated that there were several process and environmental characteristics that influenced the process yield. Data were collected on each of the variables believed to be significant, followed by regression and correlation analysis to quantify the relationships in statistical terms.

The results showed no clear correlation between anything - in spite of years of anecdotal evidence to the contrary! In fact, the underlying strong correlation between variables was confounded by excessive error in the measurement system. When the measurement systems were analyzed, many were found to exhibit error variation 2-3 times wider than the actual process spread. Measurements that were being used to control processes were often leading to adjustments that actually increased variation! People were doing their best, making things worse.

As you can see from this example, Measurement System Analysis is a critical first step that should precede any data-based decision making, including Statistical Process Control, Correlation and Regression Analysis, and Design of Experiments. The following discussion provides a broad overview of Measurement System Analysis, along with a spreadsheet analytical tool that can be downloaded (Gage R&R Worksheet) .

Before Roman units were reintroduced in 1066 by Norman William the Conqueror, there was an Anglo-Saxon (Germanic) system of measure based on the units of the barleycorn and the gyrd (rod). [ citation needed ] The systems partly merged.

Later development of the English system continued by issuing measurement standards from the then capital Winchester in about 1215. Standards were renewed in 1496, 1588 and 1758.

The last Imperial Standard Yard in bronze was made in 1845 it served as the standard in the United Kingdom until the yard was internationally redefined as 0.9144 metre in 1959 (statutory implementation: Weights and Measures Act of 1963).

Much of the units would go on to be used in later Imperial units and in the US system, which are based on the English system from the 1700s.

From May 1, 1683, King Christian V of Denmark introduced an office to oversee weights and measures, a justervæsen, to be led by Ole Rømer. The definition of the alen was set to 2 Rhine feet. Rømer later discovered that differing standards for the Rhine foot existed, and in 1698 an iron Copenhagen standard was made. A pendulum definition for the foot was first suggested by Rømer, introduced in 1820, and changed in 1835. The metric system was introduced in 1907.

### Length Edit

• mil – Danish mile. Towards the end of the 17th century, Ole Rømer connected the mile to the circumference of the earth, and defined it as 12000 alen. This definition was adopted in 1816 as the Prussian Meile. The coordinated definition from 1835 was 7.532 km. Earlier, there were many variants, the most commonplace the Sjællandsk miil of 17600 fod or 11.130 km.
• palme – Palm, for circumference, 8.86 cm
• alen – Forearm, 2 fod
• fod – Defined as a Rheinfuss 31.407 cm from 1683, before that 31.41 cm with variations.
• kvarter – Quarter, 1 ⁄ 4 alen
• tomme – Inch,
• 1 ⁄ 12 fod
• linie – Line,
• 1 ⁄ 12 tomme
• skrupel – Scruple,
• 1 ⁄ 12 linie

### Volume Edit

• potte – Pot, from 1603
• 1 ⁄ 32 foot 3
• smørtønde – Barrel of corn, from 1683 136 potter
• korntønde – Barrel of corn, from 1683 144 potter

### Miscellaneous Edit

The Dutch system was not standardised until Napoleon introduced the metric system. Different towns used measures with the same names but differing sizes.

### Weight Edit

• Ons, Once
• 1 ⁄ 16 pond = 30.881 g
• Pond (Amsterdam) – 494.09 g (other ponds were also in use)
• Scheepslast – 4000 Amsterdam pond = 1976.4 kg = 2.1786 short tons

### Length Edit

• duim –2.54 cm
• kleine palm –3 cm
• grote palm –9.6 cm, after 1820, 10 cm
• voet –12 duim = abt. 29.54 cm, many local variations
• el – about 70 cm

### Volume Edit

In Finland, approximate measures derived from body parts and were used for a long time, some being later standardised for the purpose of commerce. Some Swedish, and later some Russian units have also been used.

• vaaksa – The distance between the tips of little finger and thumb, when the fingers are fully extended.
• kyynärä – c. 60 cm – The distance from the elbow to the fingertips.
• syli – fathom, c. 180 cm – The distance between the fingertips of both hands when the arms are raised horizontally on the sides.
• virsta – 2672 m (Swedish), 1068.84 m (Russian)
• peninkulma – 10.67 km – The distance a barking dog can be heard in still air.
• poronkusema – c. 7.5 km – The distance a reindeer walks between two spots it urinates on. This unit originates from Lapland (i.e. Sápmi).
• leiviskä – 8.5004 kg
• kappa – 5.4961 l
• tynnyrinala – 4936.5 m 2 – The area (of field) that could be sown with one barrel of grain.
• kannu – 2.6172 l
• kortteli – 148 mm (length) or 0.327 l (volume)

In France, again, there were many local variants. For instance, the lieue could vary from 3.268 km in Beauce to 5.849 km in Provence. Between 1812 and 1839, many of the traditional units continued in metrified adaptations as the mesures usuelles.

In Paris, the redefinition in terms of metric units made 1 m = 443.296 ligne = 3 pied 11.296 ligne.

In Quebec, the surveys in French units were converted using the relationship 1 pied (of the French variety the same word is used for English feet as well) = 12.789 inches (of English origin). Thus a square arpent was 5299296.0804 in² or about 36,801 ft² or 0.8448 acre.

There were many local variations the metric conversions below apply to the Quebec and Paris definitions.

### Length Edit

• lieue commune – French land league, 4.452 km,
• 1 ⁄ 25 Equatorial degree
• 1 Roman cubit = 444 mm so 10000 Roman cubits = 4.44 km, a closer approximation to
• 1 ⁄ 25 degree
• lieue marine – French (late) sea league, 5.556 m, 3 nautical miles.
• lieue de poste – Legal league, 2000 toises, 3.898 m
• lieue metrique – Metric system adaptation, 4.000 m
• arpent – 30 toises or 180 pieds, 58.471 m
• toise – Fathom, 6 pieds. Originally introduced by Charlemagne in 790, it is now considered to be 1.949 m.
• pied – Foot, varied through times, the Paris pied de roi is 324.84 mm. Used by Coulomb in manuscripts relating to the inverse square law of electrostatic repulsion. Isaac Newton used the "Paris foot" in his Philosophiae Naturalis Principia Mathematica.
• pouce – Inch,
• 1 ⁄ 12 pied 27.070 mm
• ligne
• 1 ⁄ 12 pouce 2.2558 mm

### Weight Edit

Up to the introduction of the metric system, almost every town in Germany had their own definitions. It is said that by 1810, in Baden alone, there were 112 different Ellen.

### Length Edit

• Meile – 'Mile', a German geographische Meile or Gemeine deutsche Meile was defined as 7.420 km, but there were a wealth of variants:
– 7532 m – 8889 m before 1810, 8944 m before 1871, 8000 m thereafter – 7498 m – 5000 m – 7415 m, connected to a
• 1 ⁄ 15 Equatorial degree as 25406 Bavarian feet. (Prussia) – In 1816, king Frederick William III of Prussia adopted the Danish mile at 7532 m, or 24000 Prussian feet. Also known as Landmeile. – 9206 m – 9264 m – 9894 m – 5160 m – 4630 m – 4119 m – Postmeile, 7500 m. Also 9062 m or 32000 feet in Dresden – 8803 m – 11100 m, but also 9250 m – 7586 m – 1000 m – 7449 m
• ### Length Edit

• alen – Forearm, 62.748 cm from 1824, 62.75 cm from 1683, 63.26 cm from 1541. Before that, local variants.
• favn – Fathom (pl. favner), 1.882 m.
• fjerdingsvei – Quarter mile, alt. fjerding,
• 1 ⁄ 4 mil, i.e. 2.82375 km.
• fot – Foot,
• 1 ⁄ 2 alen. From 1824, 31.374 cm.
• kvarter – Quarter,
• 1 ⁄ 4 alen.
• linje – Line,
• 1 ⁄ 12 tomme or approx. 2.18 mm
• lås – 28.2 m
• landmil – Old land-mile, 11.824 km.
• mil – Norwegian mile, spelled miil prior to 1862, 18000 alen or 11.295 km. Before 1683, a mil was defined as 17600 alen or 11.13 km. The unit survives to this day, but in a metric 10 km adaptation
• rast –Lit. "rest", the old name of the mil. A suitable distance between rests when walking. Believed to be approx. 9 km before 1541.
• steinkast – Stone's throw, perhaps 25 favner, used to this day as a very approximate measure.
• stang – Rod, 5 alen or 3.1375 m
• tomme – Thumb (inch),
• 1 ⁄ 12 fot, approx. 2.61 cm. This unit was commonly used for measuring timber until the 1970s. Nowadays, the word refers invariably to the Imperialinch, 2.54 cm.
• skrupel – Scruple,
• 1 ⁄ 12 linje or approx. 0.18 mm.

### Area Edit

• mål – 100 kvadrat rode, 984 m 2 . The unit survives to this day, but in a metric 1000 m 2 adaptation.
• kvadrat rode – Square stang, 9.84 m 2
• tønneland – "Barrel of land", 4 mål

### Volume Edit

• favn – 1 alen by 1 favn by 1 favn, 2.232 m 3 , used for measuring firewood to this day.
• skjeppe
• 1 ⁄ 8 tønne, i.e. 17.4 l.
• tønne – Barrel, 139.2 l.

### Weight Edit

• bismerpund – 12 pund, 5.9808 kg
• laup – Used for butter, 17.93 kg (approx. 16.2 l). 1 laup is 3 pund or 4 spann or 72 merker.
• merke – From Roman pound, (pl. merker), 249.4 g, 218.7 g before 1683.
• ort – 0.9735 g
• pund – Pound, alt. skålpund, 2 merker 0.4984 kg, was 0.46665 kg before 1683
• skippund – Ships pound, 159.488 kg. Was defined as 151.16 kg in 1270.
• spann – Same as laup
• vette – 28.8 mark or 6.2985 kg.
• våg
• 1 ⁄ 8 skippund, 17.9424 kg.

### Nautical Edit

• favn – Fathom (pl. favner), 3 alen, 1.88 m
• kabellengde – cable length, 100 favner, 185.2 m
• kvartmil – Quarter mile, 10 kabellengder, 1852 m
• sjømil – Sea mile, 4 kvartmil, 7408 m, defined as
• 1 ⁄ 15 Equatorial degree.

### Monetary Edit

• ort – See riksdaler and speciedaler.
• riksdaler – Until 1813, Norwegian thaler. 1 riksdaler is 4 ort or 6 mark or 96 skilling.
• skilling – Shilling, see riksdaler and speciedaler.
• speciedaler – Since 1816. 1 speciedaler is 5 ort or 120 skilling. From 1876, 1 speciedaler is 4 kroner (Norwegian crown, NOK).

### Miscellaneous Edit

• tylft – 12, also dusin
• snes – 20
• stort hundre – Large hundred, 120
• gross – 144

The measures of the old Romanian system varied greatly not only between the three Romanian states (Wallachia, Moldavia, Transylvania), but sometimes also inside the same country. The origin of some of the measures are the Latin (such as iugăr unit), Slavic (such as vadră unit) and Greek (such as dram unit) and Turkish (such as palmac unit) systems.

This system is no longer in use since the adoption of the metric system in 1864.

## July 2003 AOM: Ancient Measurement Systems: Their fractional integration

From the beginning of the 20th century the study of ancient metrology has faded into the background of academic research. Before this, it was a topic of lively debate among the scientific and archaeological communities. It was considered important to clearly define the ancient modules in order to interpret architectural intentions in the ancient monuments, understand itinerary distances, the statements of the classical writers and even biblical descriptions that abound with reference to measures.

Interest in the subject was briefly revived during the 1960s by the claim of Alexander Thom who asserted that the builders of the Megalithic structures had consistently used a common unit of measure throughout the British Isles and Brittany. The claim, to this day, has been neither confirmed nor disproved. This is entirely due to the fact that there is a prevailing ignorance of the subject of ancient metrology. The statisticians who number-crunched the Megalithic data, Thom, Broadbent, Kendall, Freeman and dozens of highly qualified authors, simply failed to recognise the modules that their analyses produced. None of them had made a detailed study of ancient systems and methods of mensuration.

At first glance, the subject seems formidable, causing one learned academic to exclaim that “[Ancient] metrology is not a science, it is a nightmare.” It may seem so, there were a great many modules that were commonly used, various spans, feet, digit multiples such as pygme, remen, cubit and feet multiples — step, yard, pace, fathom, pertica and various bracchia intermediate measures of the various furlongs and stadia and the itineraries of miles, leagues, schoenus etc. However, the approach to the subject may be simplified by simply considering the basic measure of each system, which is invariably the foot. Augustus De Morgan gave a broad hint at this method of assessment in 1847, he stated:

There runs through all these national systems a certain resemblance in the measures of length and, if a bundle of rods were made of foot rules, one from every nation, ancient and modern, there would not be a very unreasonable difference in the lengths of the sticks.

It was simply by comparing the foot lengths of the different national systems that a very elegant order was perceived, leading to the conclusion that all of these “national systems” formed a single organisation. It has become customary for us to name a unit from the society in which it is found to have been used, but most often, the bureaucracies have adopted particular units during the historical period. Obviously a universal system had been fragmented into these various cultures. The difficulties of research are compounded by the lack of agreement as to nomenclature, for example, what are universally known as Roman feet are often called Attic, and at one of its variations, Pelasgo.

Which brings us to the most confusing of all the aspects of metrology — the variations. In all ancient societies, there is a quite broad variation throughout the modules, which has been wrongly regarded as slackness in the maintenance of standards. It would seem that the range of variations and the fractions by which they vary, are not merely similar from nation to nation, but identical. Once these fractions == and the simple mathematical reasons for them == had been established, it became possible to then classify these dissimilarities of the same module. The feet of the various national standards could then be compared at their correct relationships. By seeing the fractional integration through the basic foot structure, many modules could be discarded for comparative purposes, until very few Root feet remained. In fact, there are probably only twelve distinct feet from which all other “feet” are extrapolated. For example the Pythic foot is a half Saxon cubit, and many modules attributed to different cultures are in fact variations of the same basic foot, such as Saxon and Sumerian, or pied de roi and Persian. These feet in ascending order, in terms of the English foot are as follows:

Assyrian .9ft — When cubits achieve a length of 1.8ft such as the Assyrian cubit they are divisible by two, instead of the 1 ½ ft division normally associated with the cubit length. Variations of this measure are distinctively known as Oscan, Italic and Mycenaean measure. Iberian .9142857ft — This is the foot of 1/3rd of the Spanish vara, which survived as the standard of Spain from prehistory to the present. Roman .96ft — Most who are interested in metrology would consider this value to be too short as a definition of the Roman foot, but examples survive as rulers very accurately at this length. Common Egyptian .979592ft — One of the better-known measures, being six sevenths of the royal Egyptian foot. English/Greek 1ft — The English foot is one of the variations of what are accepted as Greek measure, variously called Olympian or Geographic. Common Greek 1.028571ft — This was a very widely used module recorded throughout Europe, it survived in England at least until the reforms of Edward I in 1305. It is also the half sacred Jewish cubit upon which Newton pondered and Berriman referred to as cubit A. Persian 1.05ft — Half the Persian cubit of Darius the Great. Reported in its variations throughout the Middle East, North Africa and Europe, survived as the Hashimi foot of the Arabian league and the pied de roi of the Franks. Belgic 1.071428ft — Develops into the Drusian foot or foot of the Tungri. Detectable in many Megalithic monuments. Sumerian 1.097142ft — Perhaps the most widely dispersed module of all, recorded throughout Europe, Asia and North Africa, commonly known as the Saxon or Northern foot. Yard and full hand 1.111111ft — This is the foot of the 40 inch yard widely used in mediaeval England until suppressed by statute in 1439. It is the basis of Punic measure and variables are recorded in Greek statuary from Asia Minor. Royal Egyptian 1.142857ft — The most discussed and scrutinised historical measurement. Examples of the above length are plentiful. Russian 1.166666ft — One half of the Russian arshin, one sixth of the sadzhen. Variants at one and one half of these feet as a cubit would be the Arabic black cubit, also the Egyptian cubit of the Nilometer.

Variants and variables in the above descriptions are in no wise arbitrary regional fluctuations but follow a distinct discipline. The extent of the variations covers a range of values that amounts to about one fortieth part. Immediately one can see one of the prime difficulties in the identification of ancient modules, because some of the distinct foot values are related by lesser fractions the Roman is 48 to 49 of the common Egyptian and the common Egyptian is 49 to 50 of the Greek/English. They therefore overlap at certain of their variations, in the course of comparisons this often results in the lesser variation of a distinct measure — that is essentially longer than the measure of comparison — to be shorter in length than the greater variations of the lesser measure. Metrologists continually confuse the Belgic, Frankish and Saxon/Sumerian, the latter has also been appended Ptolemaic. But, the differences become distinctively identifiable at the lengths of the pertica, chain, furlong, stadium, mile etc.

It would appear from most of the empirical evidence that the full range of the variations in a single module, here given in terms of the variations of the Greek-English foot, (the English foot being one of the series of the Greek foot) are as follows:

 Least.98867 Reciprocal.994318 Root1Standard Canonical1.0057143 Geographic1.0114612 .990916 .996578 1.002272 1.008 1.01376

The above terminology is used as descriptive in the classification of the values. It was realised from the beginning that all of these variations were impossible to express in an ascending order. They must be tabulated in two rows, the fraction linking each of the variations across the rows is 175:176, and each of the values in the top row is linked to the value directly below as 440:441. “Root” prefixes the descriptive terminology from Least to Geographic in the top row and “Standard” in the bottom row. For example, 1.008 is Standard Canonical and 1.0114612 is Root Geographic etc.

As well as these values being measurements, they are also regarded as the formulae by which any other module is classified. That is, any of the listed feet could occupy the Root position in the above table, and all of its variants would be subject to the multiplications of the tabulated values. As an example, the Persian foot when subjected to this process:

 Least1.038102 Reciprocal1.044034 Root1.05Standard Canonical1.056 Geographic1.062034 1.040461 1.046406 1.052386 1.0584 1.064448

Thus, whichever of the measures shows a direct fractional link to the English foot, such as the one and one twentieth, as above, is Root, then the maximum value of 1.064448ft is both the Hashimi foot and the original pied de roi, both could be classified as a Standard Geographic Persian foot (1.05 x 1.01376). Or the given length of the Mycenaen foot at .910315ft could be classified as a Root Geographic Assyrian foot (.9 x 1.0114612ft) and so forth. Then, whenever one is making cultural comparisons of modules, the correct classification must be selected, Root Reciprocal to Root Reciprocal etc. otherwise one is looking at a compound fraction, i.e. the fraction separating the distinctive foot plus the fraction of the variation(s), which may then show no apparent rational relationship.

These fractional separations of the rows and columns have a practical purpose they are designed to maintain integers in circular structures and artefacts such as storage and measuring vessels. If a diameter is multiple of four or a decimal, by using 22/7 as pi this results in a fractured number perimeter. Therefore 3.125 or 25/8 would be used to give an integer or rational fraction for the perimeter. Accuracy is maintained by using the longer version — by the 176th part — of the measure in the perimeter this is because 25/8 differs from 22/7 as 175 to 176. Similarly, the fraction 441 to 440 maintains integrity of number in diameter and perimeter, but of different modules. If one has a canonical perimeter number such as 360 English feet, then the diameter will be exactly 100 royal Egyptian feet, but, the royal Egyptian foot that is directly related by a ratio 8 to 7 of the English at 1.142857ft (Root), is supplanted in the diameter by the foot that is the 440th part longer at 1.145454ft (Standard). Another example is to use as a diameter 100 Standard common Greek feet, then the perimeter is 360 Assyrian feet but of the Root classification — the 440th part less. This is clearly indicative of the integrated nature of the original system, the purpose of which was the maintenance of integers in what would mostly be fractured numbers were a single standard measure used, which is what we have today. Ancient metrology is very simply based upon how number itself behaves.

## Lesson: Linear Units of Measurement Part 1

What are linear units? Why are there multiple units for linear measure?

We are going to switch gears a bit today and talk about measurement. When we talk about linear units of measurement, we will talk about customary units and metric units.

There are 2 kinds of linear measurements: we measure with customary units (inches, feet, yards, and miles) and metric units (centimeters, decimeters, meters, and kilometers). Word problem: Matt needs 36 feet of chain for a swing set. The chain is sold by the yard. How many yards of chain does he need? To solve this, I need to be able to change feet to yards. Before we do that, let’s make an anchor chart of the different units of length. Make Anchor Chart. Since yards are bigger than feet, I know my answer is going to be a smaller number. So I need to divide. I need 36 feet and there are 3 feet in each yard so I will divide 36/3 to get 12 yards.

#2How many inches are in 7 feet? Since inches are smaller, I will get a bigger number. To get a bigger number, multiply. I have 7 feet x 12 inches to give me 84 inches.
There are also metric units of length. (Add to anchor chart) Metric units are the easiest to convert because they are powers of 10. All you have to know is which is bigger or smaller. The order is millimeters, centimeters, decimeter, meter and kilometer.

#3. How many meters are in 400 centimeters? If I have 400 centimeters, and I know that there are 100 centimeters in a meter, I can divide to get 4.

Let’s try the first few conversions on your sheet together. #1-4: Guided Practice

Now you will complete the assignment on your own

Journal: How do you know whether to multiply or divide to convert units?

How are metric units different from customary units?

Foot, yard, inch, mile, centimeter, millimeter, decimeter, meter, kilometer

## Spartan Ideas

So this is the final installment of the “On quantum measurement” series. You may have arrived here by reading all previous parts in one sitting (I’ve heard of such feats in the comments). This is the apotheosis: what all these posts have been gearing up to. If, for some reason that only the Internets know, you have arrived here without the benefit of the first six installments, I’ll provide you with the link to the very first installment, but I won’t summarize all the posts, out of deference to all the readers who got here the conventional way.

The Copenhagen Interpretation of quantum mechanics, as I’m sure all of you that have arrived to Part 7 are aware of, is a view of the meaning of quantum mechanics promulgated mostly by the Danish physicist Niels Bohr, and codified in the 1920s, that is, the “heydays” of quantum physics. Quantum mechanics can be baffling to be sure, and there are multiple attempts to square what we observe experimentally with our common sense. The Copenhagen Interpretation is an extreme view (in my opinion) of how to make sense of the reflection of the quantum world in our classical measurement devices. So, at its very core, the Copenhagen Interpretation muses about the relationship of the classical to the quantum world.

As a young student of quantum mechanics in the early eighties, I was a bit baffled by this right away. When the true underlying physics is quantum (I mused), and that therefore the classical world is just an approximation of the quantum, how can we have “theorems” that codify the relationship between quantum and classical systems?

I won’t write a treatise here about the Copenhagen Interpretation. I’ve already linked the Wikipedia article about it, which should get those of you who are not yet groaning up to speed. I’ll just list the two central “things” that are taught just about everywhere quantum mechanics is taught, and that can be squarely traced back to Bohr’s school.

1. Physical systems do not have definite properties prior to being measured, but instead should be described by a set of probabilities
2. The act of measurement changes the quantum system, so that it takes on only one of the previous possibilities (wave function collapse, or reduction)

Yes, the general understanding of the Copenhagen Interpretation is more multi-faceted, but for the purpose of this post I will focus on the collapse of the wave function. When I first fully understood what that meant, it was immediately clear to me that this was just a load of crap. I knew of no law of physics that could engender such a collapse, and it violated everything I believed in (such as conservation of probabilities). You who reads this blog so ardently already know this: it makes no sense from the point of view of information theory.

Now, quantum information theory did not exist around the time of Bohr (and Heisenberg, who must carry some of the blame for the Copenhagen Interpretation). And maybe the two should get a pass for this simple reason, except for the fact that John von Neumann, as I have pointed out in another post), had the foundations of quantum information theory already worked out in 1932, two years after the first “definitive” treatise on the “Copenhagen spirit” was published by Heisenberg.

So you, faithful reader, come to this post well prepared. You already know that Hans Bethe told me and my colleague Nicolas Cerf that we showed that wave functions don’t collapse, you know that John von Neumann almost discovered quantum information theory in the 30s, that quantum measurement is very different from its classical counterpart because copying is not allowed in the quantum world. You know where Born’s rule comes from, and you pondered the utility of quantum Venn diagrams. You were promised a discussion of Schrödinger’s cat, but that never materialized. Instead, you were given a discussion of the quantum eraser. Arguably, that is a more interesting system, but I understand if you are miffed. But to make it up, now we get to the quantum grand-daddy of them all. I will show you that the Copenhagen interpretation is not only toast theoretically, but that it is possible to design experiments that will show this. Or they will show that I’m full of the aforementioned crap. Either way, it is going to be exciting.

In this post, I will reveal to you the mathematical beauty and elegance of consecutive measurements performed on the same quantum system. I will also show you how looking at three measurements in a row (but not two), will reveal to you that the Copenhagen Interpretation is now history, ripe for the trash heap of ill-conceived concepts in theoretical physics. All of what I’m going to tell you is an extension of the picture that Nicolas Cerf and I wrote about in 1996, and which Bethe understood immediately after we showed him our results, while it took us six months to understand what he told us. But it is an extension that took some time to clarify, so that the indictment of Bohr (and implicitly Heisenberg) and the collapse picture of measurement is unambiguous, and most importantly, experimentally verifiable.

Let’s get right into the thick of things. But getting started may really be the hardest thing here. Say you want to measure a quantum system. But you know absolutely nothing about it. How do you write such a quantum system?

In general, people write arbitrary quantum states like this: (|Q angle=sum_ialpha_i |i angle), with complex coefficients αi that satisfy (|Q angle=sum_ialpha_i |i angle). But you may ask, “Who told you what basis to write this quantum state in? The basis states (alpha_i), I mean”. After all, the amplitudes (alpha_i) only make sense with respect to a particular basis system (if you transform this basis to another, as we will do a lot in this post) it changes the coefficients. “So haven’t you already assumed a lot by writing the quantum state like that?” (You may remember questions like that from a blog post on classical information, and this is no accident).

If you think about this problem for a little while, you realize that indeed the coefficients and the basis you choose are crucial. Just as in classical information theory where I told you that the entropy of a system was undefined, and determined only by the measurement device that you were about to use to learn about it, the state of an arbitrary quantum system only makes sense relative to the quantum states of the detector that you are about to use to measure it. This is, essentially, what is at the heart of the “relative state” formalism of quantum mechanics, due to Everett, of course. That fellow Hugh Everett does not get as much recognition as he deserves, so I’ll let you gaze at him for a little while.

 H. Everitt III (1930-1982) Source: Wikimedia

He cooked up his theory as a graduate student, but as nobody believed his theory at the time, he left quantum physics and became a defense analyst.

You may expect me to launch into a description and discussion of the “many-worlds” interpretation of quantum mechanics, which became a fad in the 1970s, but I won’t. It is silly to call the relative-state picture a “many-worlds” interpretation, because it does not propose at all that at every quantum measurement event the universe splits into so many worlds as there are orthogonal states. This is beyond silly in fact (it was also not at all advocated by Everett), and the people who did coin these terms should be ashamed of themselves (but I won’t name them here). My re-statement of Everett’s theory in the modern language of quantum information theory can be read here, and in any case Zeh (in 1973) and Deutsch (in 1985) before me had understood much about Everett’s theory without imagining some many-worlds voodoo.

So let us indeed talk about a quantum state by writing it in terms of the basis states of the measurement device we are about to examine it with. Because that is all we can do, ever. Just as we have learned in the first six installments of this series, we will measure the quantum state using an ancilla A, with orthogonal basis states (|i angle_A) I wrote the ‘A’ as a subscript to distinguish it from the quantum states, but later I will drop the subscript once you are used to the notation.

Now look what happens if I measure (|Q angle=sum_ialpha_i |a_i angle) with A (to distinguish the quantum states, written in terms of A’s basis from the A Hilbert space, we simply write them as (|a_i angle)). The probability to observe the quantum state in state i is (you remember of course Part 4)

Now get this: You’re supposed to measure a random state, but the probability distribution you obtain is not random at all, but given by the probability distribution pi, which is not uniform. This makes no sense at all. If (|Q angle) was truly arbitrary, then on average you should see (p_i=1/d) (the uniform distribution), where d is the dimension of the Hilbert space. So an arbitrary unknown quantum state, written in terms of the basis states of the apparatus that we are going to measure it in, should be (and must be) written as

Now, each outcome i is equally likely, as it should be if you are measuring a state that nobody prepared beforehand. A random state. With maximum entropy.

So now we got this out of the way: We know how to write the to-be-measured state. Except that we assumed that the system Q had never interacted with anything (or was measured by anything) before. This also is a nonsense assumption. All quantum states are entangled: there is no such thing as a “pristine” quantum system. Fortunately, we know exactly how to describe that: we can write the quantum wavefunction so that it is entangled with an arbitrary “reference” state R:

You can think of R as all the measurement devices that Q has interacted with in the past: who are we to say that A is really the first? Now we don’t know really what all these R states are, so we just trace them out, so that the Q density matrix is the familiar

( ho_Q=frac1dsum_i |a_i anglelangle a_i|.)

After we measured the state with A, the joint state QRA is now (the previous posts tell you how to do this)

Don’t worry about the R system too much: the Q density matrix is still the same as above, and I have to skip the reason for that here. You can read about it in the paper. Oh yes, there is a paper. Read on.

This is, after all, the post about consecutive measurements, so we will measure Q again, but this time with ancilla B, which is not in the same basis as A. (If it was, then the result would be trivial: you’d just get the same result over and over again: it is like all the pieces of the measurement device A all agreeing on the result).

So we will say that the B eigenstates are at an angle with the A eigenstates:

This just means that what is a zero or one in one of the measurement devices (if we are measuring qubits) is going to be a superposition in the other’s basis. U is a unitary matrix. For qubits, a typical U will look like this:

where θ is the angle between the bases. (Yes, it is a special case, but it will suffice.)

To measure Q with B (after we measured it with A, of course) we have to write Q in terms of B’s eigenstates, and then measure. What you get is a wave function that has Q entangled not only with its past (R), but both A and B as well:

You might think that this looks crazy complicated, but the result is really quite simple. And it agrees with everything that has been written about consecutive measurements so far, whether they advocated a collapse picture or a unitary “relative state” picture. For example, the joint density matrix of just the two detectors, ( ho_), is just

( ho_=frac1dsum_i|i anglelangle i|otimessum_j|U_|^2|j anglelangle j |.)

That this is the “standard” result will dawn on you when you notice that (|U_|^2) is the conditional probability to measure outcome j with B given that the previous measurement (with A) gave you outcome i (with probability (1/d), of course).

It is fair warning that if you have not understood this result, you should probably not go on reading. Go on if you must, but remember to go back to this result.

Also, keep in mind that I will from now on use the index i for the system A, the index j for system B, and later on I will use k for system C. And I won’t continually indicate the state with a bothersome subscript like (|i angle_A). Because that is how I roll.

So here is what we have achieved. We have written the physics of consecutive quantum measurements performed on the same system in a manifestly unitary formalism, where wavefunctions do not collapse, and the joint wavefunction of the quantum system, entangled with all the measurements that have preceded our measurements, along with our recent attempts with A and B, exists in a superposition, will all the possibilities (realized or not) still present. And the resulting density matrix along with all the probabilities agree precisely with what has been known since Bohr, give or take.

And the whispers of “Chris, what other ways do you know to waste your time, besides I mean, blogging?” are getting louder.

But wait. There is the measurement with C that I advertised. You might think (possibly with anybody who has ever contemplated this calculation) “Why would things change?” But they will. The third measurement will show a dramatic difference, and once we’re done you’ll know why.

First, we do the boring math. You could do this yourself (given that you followed enough to get to be able to derive Eqs. (1) and (2). You just use a unitary U′ to encode the angle between the measurement system C and the system B (just like U described the rotation between systems A and B), and the result (after tracing out the quantum system Q and the reference system R, since no one is looking at those) looks innocuous enough:

Except after looking this formula over a couple of times, you squint. And then you go “Hold on, hold on”.

“The B measurement!”, you exhale. After measuring with B the device was diagonal in the measurement basis (this means that the density matrix was like (|j angle langle j|)). But now you measured Q again, and now B is not diagonal anymore (now it’s like (|j angle langle j’|)). How is that possible?

Well, it is the law, is all I can tell you. Quantum mechanics requires it. Density matrices, after all, only tell us part of the story (since you are tracing out the entire history of measurements). That story could be full of lies, and here it turns out it actually is.

It is the last measurement that gives a density matrix that is diagonal in the measurements basis, always. Oh, and the first one, if you measure an arbitrary unknown state. That’s two. To see that things can be different, you need a third. The one in between.

To see that Eq. (3) is nothing like what you are used to, let’s see what a collapse picture would give you. A detailed calculation using the conventional formalism will lead to (the superscript “coll”) is to remind you that this NOT the result of a unitary calculation

( ho_^<< m coll>>=frac1dsum_i|i anglelangle i|otimes sum_j|U_|^2|j anglelangle j|otimes sum_k|U’_|^2|k anglelangle k|.)(4)

The difference between (3) and (4) should be immediately obvious to you. You get (4) from (3) if you set j=j′, that is, if you remove the off-diagonal terms that exist in (3). But, you see, there is no law of physics that allows you to just grab some off-diagonal terms and yank them out of the matrix. That means that (3) is a consequence of quantum mechanics, and (4) is not derived from anything. It is really just wishful thinking.

“So”, I can hear you mutter from a distance, “can you make a measurement that supports one or the other of the approaches? Can experiments tell the difference between the two ways to understand quantum measurement?”

That, Detective, is the right question.

How do we tell the difference between two density matrices? Let us focus on qubits here (d=2). And, just to make things more tangible, let’s fix the angles between the consecutive measurements.

Measurement A is the first measurement, so there is no angle. In fact, A sets the stage and all subsequent measurements will be relative to that. We will take B at 45 degrees to A. This means that B will have a 50/50 chance to record 0 or 1, no matter whether A registered 0 or 1. Note that A also will record 0 or 1 half the time, as it should in the initial state is random and unknown.

We will take C to measure at an angle of 45 degrees to B also, so that C’s entropy will be one bit as well. Thus, all three detector’s entropy should be one bit. This will be true, by the way, both in the unitary, and in the collapse picture. The relative states between the three detectors are, however, quite different between the two descriptions. Below you can see the quantum Venn diagram for the unitary picture on the left, and the collapse picture on the right.

We kinda knew that had to be like that, on account of the π/4 angles and all. Yes, the two diagrams look very different. For example, look at detector B. If I give you A and C, the state of B is perfectly known as S(B|AC)=0). That’s not true in the collapse picture: giving A or C does nothing for B.

That in itself looks like a death knell for the unitary picture: How could it be that a past and a future experiment can fully determine the quantum state in the present? It turns out that such questions have been asked before! Aharonov, Bergmann, and Lebowitz (ABL) showed in 1964 that it is possible to set up a measurement so that knowing the results from A and C will allow you to predict with certainty what B would have recorded [1]. As you can tell from the title of their paper, ABL were concerned about the apparent asymmetry in quantum measurement.

Of course there is an asymmetry! a measurement can tell you about the past, but it cannot tell you about the future! What an asymmetry!

Slow down, there. That’s not a fair comparison. Causality is, after all, ruling over us all: what hasn’t happened is different from that which has happened. The real question is whether, after all things are said and done, there is an asymmetry between what was, and what could have been. In the language of quantum measurement, we should instead ask the question: If the past measurements influence what I can record in the future, do the future measurements constrain what once was, in an equal manner? Or put in another way, can can the measurements today tell me as much about the state on which it was performed, as knowing the state today tells you about future measurements?

To some extent, ABL answered this question in the affirmative. For a fairly contrived measurement scenario, they showed that if you give me the measurement record of the past, as well as what was measured in the future, I can tell you what it is you must have measured in the present. In other words, they said that the past and future, taken together, will predict the present perfectly.

I don’t think everybody who read that paper in 1964 was aware of the ramifications of this discovery. I don’t think people are now. What we show in our paper is that what ABL showed holds in a fairly contrived situation, in fact holds true universally, all the time.

Can’t you wait just a little longer? I promise it will be at the end of the blog. You can scroll ahead if you must.

In fact, we show that the ABL result is just a special case that holds quite generally. For any sequence of measurements of the same quantum system, Jennifer Glick and I prove that only the very first and the very last measurements are uncertain. All those measurements in between are perfectly predictable. (This holds for the case of measuring unprepared quantum states only.) This makes sense from the point of view I just advocated: you cannot fully know the last measurement because the future did not yet happen. And you cannot know the first measurement because there is nothing in its past. Everything else is perfectly knowable.

Now, “knowable” does not mean “known”, because in general you cannot use the results of the individual measurements to make the predictions about the intermediate detectors: you need some of the off-diagonal terms of the density matrix, which means that you have to perform more complex, joint measurements. But you only need the measurement devices, nothing else.

We show a number of other fairly uncommon things for sequences of quantum measurements in the paper aptly entitled “Quantum mechanics of consecutive measurements”, which you can read on arXiv here. For example, we show that the sequence of measurements does not form a Markov chain, as is expected for a collapse picture. We also show that the density matrix of any pair of detectors in that sequential chain is “classical”, which we here identify with “diagonal in the detector product basis”. There are several more general results in there: be sure to read the Supplementary Material, where all the proofs reside.

“So your math says that wavefunctions don’t collapse. Can you prove it experimentally?”

That too is an excellent question. Math, after all, it just a surrogate that helps us understand the laws of nature. What we are saying is that the laws of nature are not as you thought they were. And if you make a statement like that, then it should be falsifiable. If your theory truly goes beyond the accepted canon, then there must be an experiment that will support the new theory (it cannot prove it, mind you) by sending the old theory to where…. old theories go to die.

What is that experiment? It turns out it is not an easy experiment. Or, at least, for this particular scenario (three consecutive measurements of the same quantum system) the experiment is not easy. The statistics of counts of the three measurement devices is predicted by the diagonal of the joint density matrix ( ho_), and this is the same in the unitary relative state picture and the collapse picture. The difference is in off-diagonal elements of the density matrix. Now, there are methods that allow you to measure off-diagonal elements of a quantum state, using so-called “quantum-state tomography” methods. Because the density matrix in question is large (an 8࡮ matrix for qubit measurements), this is a very involved measurement. Fortunately, there are short cuts. It turns out that for the case at hand, every single moment of the density matrix is different. The nth moments of a density matrix is defined by ( < m Tr> ho^n), and it turns out that already the second moment, that is ( < m Tr> ho^2) is different. Measuring the second moment of the density matrix is far simpler than measuring the entire matrix via quantum state tomography, but given that it is a three qubit system, it is still not a simple endeavor. But it is one that I hope someone will be convinced will be worth undertaking. Because it will the experiment that will send the Copenhagen interpretation packing, for all time.

So I asked myself, “How do I close such a long series about quantum measurement, and this interminable last post?” I hope to have brought quantum measurement a little bit out of the obscure corner where it is sometimes relegated to. Much about quantum measurement can be readily understood, and what mysteries there still are can, I am confident, be resolved as well. Collapse never made any physical sense to begin with, but neither did a branching of the universe. We know that quantum mechanics is unitary, and we now know that the chain of measurements is too. What remains to be solved, really, is just the randomness that we experience in the last measurement, when the future is still uncertain.

Where does this randomness come from? What do these probabilities mean? I have some ideas about that, but this will have to wait for another blog post. Or series.

[1] Y. Aharonov, P. G. Bergmann and J. L. Lebowitz, “Time symmetry in the quantum process of measurement,” Phys. Rev. B 134, 1410–16 (1964).

## Face Validity

Face validity is the extent to which a measurement method appears “on its face” to measure the construct of interest. Most people would expect a self-esteem questionnaire to include items about whether they see themselves as a person of worth and whether they think they have good qualities. So a questionnaire that included these kinds of items would have good face validity. The finger-length method of measuring self-esteem, on the other hand, seems to have nothing to do with self-esteem and therefore has poor face validity. Although face validity can be assessed quantitatively—for example, by having a large sample of people rate a measure in terms of whether it appears to measure what it is intended to—it is usually assessed informally.

Face validity is at best a very weak kind of evidence that a measurement method is measuring what it is supposed to. One reason is that it is based on people’s intuitions about human behaviour, which are frequently wrong. It is also the case that many established measures in psychology work quite well despite lacking face validity. The Minnesota Multiphasic Personality Inventory-2 (MMPI-2) measures many personality characteristics and disorders by having people decide whether each of over 567 different statements applies to them—where many of the statements do not have any obvious relationship to the construct that they measure. For example, the items “I enjoy detective or mystery stories” and “The sight of blood doesn’t frighten me or make me sick” both measure the suppression of aggression. In this case, it is not the participants’ literal answers to these questions that are of interest, but rather whether the pattern of the participants’ responses to a series of questions matches those of individuals who tend to suppress their aggression.

## OTHER WORDS FROM measurement

As measurement s grow more precise, the approximation schemes theorists use to make predictions may not be able to keep up.

USGS used laser-equipped drones to scan San Diego’s land topography paired with ocean measurement s to produce these maps.

Governments around the world have set different prices based on how harmful they believe a metric ton of carbon is to society, a highly debatable measurement .

It provides a way of measuring area in the space and allows you to change the space’s shape only if area measurement s stay constant.

Most convincingly, in 2015, one team overlaid many measurement s of blazars behind voids and managed to tease out a faint halo of low-energy gamma rays around the blazars.

Thinking of longer journeys to Mars or an asteroid would require careful measurement of conditions.

No one in the Hall of Fame was the very best at every hitting or fielding measurement all the time either.

These days it appears as if women will go to great lengths to achieve that 36-24-36 measurement .

Precise measurement of the forces on the head after a tackle is critical.

Gentlemen, imagine, if you will, having your most intimate measurement read out in open court?

At last, however, the four men came together, and proceeded to the measurement of swords.

Of course, you understand that measurement of anything is the comparing of it with some established standard.

The only known standard for the measurement of time is the movement of the earth in relation to the stars.

You might make it a rule to be away from your subject a distance of about three or four times the extreme measurement of it.

The so-called measurement of them was even preserved in families, and was reputed to be a charm.

## 7.6: Systems of Measurement (Part 1)

· Describe the general relationship between the U.S. customary units and metric units of length, weight/mass, and volume.

· Define the metric prefixes and use them to perform basic conversions among metric units.

In the United States, both the U.S. customary measurement system and the metric system are used, especially in medical, scientific, and technical fields. In most other countries, the metric system is the primary system of measurement. If you travel to other countries, you will see that road signs list distances in kilometers and milk is sold in liters. People in many countries use words like “kilometer,” “liter,” and “milligram” to measure the length, volume, and weight of different objects. These measurement units are part of the metric system.

Unlike the U.S. customary system of measurement, the metric system is based on 10s. For example, a liter is 10 times larger than a deciliter, and a centigram is 10 times larger than a milligram. This idea of “10” is not present in the U.S. customary system—there are 12 inches in a foot, and 3 feet in a yard…and 5,280 feet in a mile!

So, what if you have to find out how many milligrams are in a decigram? Or, what if you want to convert meters to kilometers? Understanding how the metric system works is a good start.

The metric system uses units such as meter, liter, and gram to measure length, liquid volume, and mass, just as the U.S. customary system uses feet, quarts, and ounces to measure these.

In addition to the difference in the basic units, the metric system is based on 10s, and different measures for length include kilometer, meter, decimeter, centimeter, and millimeter. Notice that the word “meter” is part of all of these units.

The metric system also applies the idea that units within the system get larger or smaller by a power of 10. This means that a meter is 100 times larger than a centimeter, and a kilogram is 1,000 times heavier than a gram. You will explore this idea a bit later. For now, notice how this idea of “getting bigger or smaller by 10” is very different than the relationship between units in the U.S. customary system, where 3 feet equals 1 yard, and 16 ounces equals 1 pound.

Length, Mass, and Volume

The table below shows the basic units of the metric system. Note that the names of all metric units follow from these three basic units.

other units you may see

In the metric system, the basic unit of length is the meter. A meter is slightly larger than a yardstick, or just over three feet.

The basic metric unit of mass is the gram. A regular-sized paperclip has a mass of about 1 gram.

Among scientists, one gram is defined as the mass of water that would fill a 1-centimeter cube. You may notice that the word “mass” is used here instead of “weight.” In the sciences and technical fields, a distinction is made between weight and mass. Weight is a measure of the pull of gravity on an object. For this reason, an object’s weight would be different if it was weighed on Earth or on the moon because of the difference in the gravitational forces. However, the object’s mass would remain the same in both places because mass measures the amount of substance in an object. As long as you are planning on only measuring objects on Earth, you can use mass/weight fairly interchangeably—but it is worth noting that there is a difference!

Finally, the basic metric unit of volume is the liter. A liter is slightly larger than a quart.

The handle of a shovel is about 1 meter.

A paperclip weighs about 1 gram.

A medium-sized container of milk is about 1 liter.

Though it is rarely necessary to convert between the customary and metric systems, sometimes it helps to have a mental image of how large or small some units are. The table below shows the relationship between some common units in both systems.

Common Measurements in Customary and Metric Systems

1 centimeter is a little less than half an inch.

1.6 kilometers is about 1 mile.

1 meter is about 3 inches longer than 1 yard.

1 kilogram is a little more than 2 pounds.

28 grams is about the same as 1 ounce.

1 liter is a little more than 1 quart.

4 liters is a little more than 1 gallon.

Prefixes in the Metric System

The metric system is a base 10 system. This means that each successive unit is 10 times larger than the previous one.

The names of metric units are formed by adding a prefix to the basic unit of measurement. To tell how large or small a unit is, you look at the prefix. To tell whether the unit is measuring length, mass, or volume, you look at the base.

Prefixes in the Metric System

1,000 times larger than base unit

100 times larger than base unit

10 times larger than base unit

10 times smaller than base unit

100 times smaller than base unit

1,000 times smaller than base unit

Using this table as a reference, you can see the following:

· A kilogram is 1,000 times larger than one gram (so 1 kilogram = 1,000 grams).

· A centimeter is 100 times smaller than one meter (so 1 meter = 100 centimeters).

· A dekaliter is 10 times larger than one liter (so 1 dekaliter = 10 liters).

Here is a similar table that just shows the metric units of measurement for mass, along with their size relative to 1 gram (the base unit). The common abbreviations for these metric units have been included as well.

## Looking for better customer relationships?

Instead of putting all that effort into delighting the customer, the authors argue it should be invested in making the customer experience and problem resolution as easy as possible .

The authors found that the the ease of having your problems resolved was a much better predictor for satisfaction than having expectations exceeded. Improve the customer experience by easifying your customer’s journey.

This new service philosophy requires different measurements, which is why the CES was developed. They showed that CES is superior to CSAT and NPS in predicting consumer behavior .

Don’t ask, “How satisfied are you with this service?” ask, “How easy was it to get in contact/make a purchase/have your issue resolved?”

Relevance is crucial here. The time to pop the question is right after your customer had her experience. Otherwise, the ease of the experience might have been forgotten. It can be asked in-app (ease of the website/app experience), via live chat, or via email (ease of the service).

CheckMarket offers a free template to create your own CES survey. With some tweaking, many customer service tools are suited for this purpose. Read more in our post on how to get to the right customer effort score question .

Social media has had an immense impact on the relationship between business and customer. Where before, a great or poor service experience would maybe be shared with the closest family and friends, social media offered an outlet and reach to potentially millions of people.

Because of that, it’s the perfect place to hear what your customers are really thinking about you. If you have the right tools to track this, that is.

Facebook and Twitter are of course relevant platforms to track, but also platforms like Quora, Yelp, TripAdvisor, etc.

• Mention . A powerful freemium tool that gives you a heads up whenever your brand is mentioned on the web. It’s especially handy for social media tracking, for which Google Alerts is not suitable.
• Socialmention . A free tool that analyzes social mentions of your brand on the web. Among others, it shows the likeliness of your brand being discussed on the web, the ratio of positive to negative mentions, the likelihood of people mentioning your brand repeatedly and the range of influence.

This metric originates from the Lean Six Sigma approach , and measures the number of complaints, or "Things Gone Wrong," per 100, 1000, or up to a 1,000,000 units of survey responses, units sold, or other.

The standard approach to measure TGW is through complaint sections in customer surveys, but you could also maintain internal metrics. In the worst case scenario your score is 1 or higher, meaning that you get at least 1 complaint per chosen unit.

Once you start measuring, you can start optimizing. And optimizing your customer satisfaction is the best investment you can make.

For related topics on measurement methods, check out our posts on measuring customer loyalty , how to measure service quality , and customer service KPIs .

Pascal is Mr. Marketing at Userlike. Besides leading Userlike’s marketing plan for world domination, he fills his days watching old movies.