# 4: Fractions

Often in life, whole amounts are not exactly what we need. A baker must use a little more than a cup of milk or part of a teaspoon of sugar. Similarly a carpenter might need less than a foot of wood and a painter might use part of a gallon of paint. In this chapter, we will learn about numbers that describe parts of a whole. These numbers, called fractions, are very useful both in algebra and in everyday life. You will discover that you are already familiar with many examples of fractions!

• 4.1: Visualize Fractions (Part 1)
A fraction is a way to represent parts of a whole. The denominator b represents the number of equal parts the whole has been divided into, and the numerator a represents how many parts are included. The denominator, b, cannot equal zero because division by zero is undefined. A mixed number consists of a whole number and a fraction. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one.
• 4.2: Visualize Fractions (Part 2)
Equivalent fractions are fractions that have the same value. When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. We can use the inequality symbols to order fractions. Remember that a > b means that a is to the right of b on the number line. As we move from left to right on a number line, the values increase.
• 4.3: Multiply and Divide Fractions (Part 1)
A fraction is considered simplified if there are no common factors, other than 1, in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.
• 4.4: Multiply and Divide Fractions (Part 2)
The reciprocal of the fraction a/b is b/a, where a ≠ 0 and b ≠ 0. A number and its reciprocal have a product of 1. To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator. To divide fractions, multiply the first fraction by the reciprocal of the second.
• 4.5: Multiply and Divide Mixed Numbers and Complex Fractions (Part 1)
To multiply or divide mixed numbers, convert the mixed numbers to improper fractions. Then follow the rules for fraction multiplication or division and then simplify if possible. A complex fraction is a fraction in which the number and/or denominator contains a fraction. To simplify a complex fraction, rewrite the complex fraction as a division problem. Then follow the rules for dividing fractions and then simplify if possible.
• 4.6: Multiply and Divide Mixed Numbers and Complex Fractions (Part 2)
Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. When the numerator and denominator have different signs, the quotient is negative. If both the numerator and denominator are negative, then the fraction is positive because we are dividing a negative by a negative. Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses.
• 4.7: Add and Subtract Fractions with Common Denominators
To add fractions, add the numerators and place the sum over the common denominator. To subtract fractions, subtract the numerators and place the difference over the common denominator.
• 4.8: Add and Subtract Fractions with Different Denominators (Part 1)
The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators. To find the LCD of two fractions, factor each denominator into its primes. Then list the primes, matching primes in columns when possible, and bring down the columns. Finally, multiply the factors together, the product is the LCM of the denominators which is also the LCD of the fractions.
• 4.9: Add and Subtract Fractions with Different Denominators (Part 2)
In fraction multiplication, you multiply the numerators and denominators together, respectively. To divide fractions, you multiply the first fraction by the reciprocal of the second. For fraction addition, add the numerators together and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD. Likewise, for fraction subtraction, subtract the numerators and place the difference over the common denominator.
• 4.10: Add and Subtract Mixed Numbers (Part 1)
To add mixed numbers with a common denominator, first rewrite the problem in vertical form. Then, add the whole numbers and the fractions together. Finally, simplify the sum if possible. An alternate method for adding mixed numbers is to convert the mixed numbers to improper fractions and then add the improper fractions. This method is usually written horizontally.
• 4.11: Add and Subtract Mixed Numbers (Part 2)
To subtract mixed numbers with common denominators, first rewrite the problem in vertical form and compare the two fractions. If the top fraction is larger than the bottom fraction, subtract the fractions and then the whole numbers. If the top fraction is not larger than the bottom fraction, in the top mixed number, take one whole and add it to the fraction part, making a mixed number with an improper fraction. Then subtract the fractions and then the whole numbers. Lastly, simplify if possible.
• 4.12: Solve Equations with Fractions (Part 1)
The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, or a fraction. To determine whether a number is a solution to an equation, first substitute the number for the variable in the equation. Then simplify the expressions on both sides of the equation and determine if the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.
• 4.13: Solve Equations with Fractions (Part 2)
To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
• 4.14: Fractions (Exercises)
• 4.15: Fractions (Summary)

Figure 4.1 - Bakers combine ingredients to make delicious breads and pastries. (credit: Agustín Ruiz, Flickr)

## 4: Fractions

Simple number or constant.

Action that involves two numbers (i.e. 6+2).

Action that requires one number (i.e. 40%).

Actions about the calculator or the tape.

Number formating actions fractions and scientific notation.

### Keyboard

You can use your numeric keypad to insert numbers along with the keys 'enter', 'equals', 'backspace', 'delete', as well as the + - * / keys.

### Tape and Tape Buttons

All calculations are saved on the tape. Click on any number or operator on the tape and change it at any time. Hit equals and the new result will appear.

You can use the print button to print out the tape.

### Clearing Buttons

Clear button clears the last input.

All clear button clears the calculator, tape, and resets any functions.

Memory clear button clears the memory.

### Memory Buttons

Memory recall button retrieves the number you have in memory and places it in the display field.

Memory plus button adds the number displayed to the contents of the memory.

Memory minus button subtracts the number displayed from the contents of the memory.

### Function Buttons

% Percent button is used to find the percentage of a number. Enter the percentage amount, click the % button, then enter the number you want the percentage of, and then click equals. i.e. 20% 125 = 25 where 25 is 20% of 125. Note: The percent function will also work if you enter the number first and then the percentage you want i.e. 125 %20 = 25.

## 4: Fractions

Note that the whole number-integral part is: empty
The decimal part is: .4 = 4 /10
Full simple fraction breakdown: 40/100
= 4/10
= 2/5

Scroll down to customize the precision point enabling 0.4 to be broken down to a specific number of digits. The page also includes 2-3D graphical representations of 0.4 as a fraction, the different types of fractions, and what type of fraction 0.4 is when converted.

##### Level of Precision for 0.4 as a Fraction

The level of precision are the number of digits to round to. Select a lower precision point below to break decimal 0.4 down further in fraction form. The default precision point is 5.

If the last trailing digit is ř" you can use the "round half up" and "round half down" options to round that digit up or down when you change the precision point.

For example 0.875 with a precision point of 2 rounded half up = 88/100, rounded half down = 87/100.

##### Graph Representation of 0.4 as a Fraction

Pie chart representation of the fractional part of 0.4

2D Chart 3D Chart

##### Numerator & Denominator for 0.4 as a Fraction

0.4 = 0 4 /10
numerator/denominator = 4 /10

##### Is 4 /10 a Mixed, Whole Number or Proper fraction?

A mixed number is made up of a whole number (whole numbers have no fractional or decimal part) and a proper fraction part (a fraction where the numerator (the top number) is less than the denominator (the bottom number). In this case the whole number value is empty and the proper fraction value is 4/10.

##### Can all decimals be converted into a fraction?

Not all decimals can be converted into a fraction. There are 3 basic types which include:

Terminating decimals have a limited number of digits after the decimal point.

Example: 8303.71 = 8303 71 /100

Recurring decimals have one or more repeating numbers after the decimal point which continue on infinitely.

Example: 4553.3333 = 4553 3333 /10000 = 333 /1000 = 33 /100 = 1 /3 (rounded)

Irrational decimals go on forever and never form a repeating pattern. This type of decimal cannot be expressed as a fraction.

## How to Do Fractions

This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.

There are 12 references cited in this article, which can be found at the bottom of the page.

Fractions represent how many parts of a whole you have, which makes them useful for taking measurements or calculating precise values. Fractions can be a difficult concept to learn since they have special terms and rules for using them in equations. Once you understand the parts of a fraction, practice doing addition and subtraction problems with them. When you know how to add and subtract fractions, you can move on to trying multiplication and division with fractions.

## Fractions Resources

Numerator and Denominator
The numerator is the top number in the fraction and is the number of parts used. The denominator is the bottom number in the fraction and is the number of parts that make up a whole. For example, if we are looking at a pizza and we are told that someone ate 2 &frasl8 of the pizza, the numerator would be 2 (the number of slices eaten) and the denominator would be 8 (because there are 8 pieces total).

Equivalent Fractions
Equivalent fractions are fractions that have different numbers as the numerator and denominator, but are actually the same. For example, 4 &frasl8 = 3 &frasl6 = 2 &frasl4 = 1 &frasl2.

Proper Fractions vs Improper Fractions
Proper fractions are any fractions where the numerator is less than the denominator. 8 &frasl9 and 2 &frasl3 are both proper fractions. Improper fractions are any fractions where the numerator is greater than or equal to the denominator. 9 &frasl4 and 5 &frasl5 are both improper fractions.

Mixed Fractions
Mixed fractions are used to show when there is a whole plus a part involved. For example, if someone ate 2 whole pizzas and 1 &frasl2 of another pizza, the mixed fraction of how many pizzas they ate would be equal to 2 1 &frasl2. Mixed fractions can be converted to improper fractions by multiplying the whole number by the denominator, adding the numerator to their product, and putting that sum over the original denominator. Similarly, improper fractions can be converted to mixed fractions by dividing the numerator by the denominator to get the whole number and using the remainder as the new numerator.

Now that you have a better understanding of the different types of fractions your child will be working with, scroll up to check out our fraction worksheets and exercises!

## Fraction Worksheets

Several different types of fraction worksheets are available on the pages below. Includes basic fraction worksheets, equivalent fractions, comparing fractions, ordering fractions, and more.

Printable fraction games and printable worksheets Manipulative fraction strips, printable fraction pizzas, a memory-matching game, and more.

This page has worksheets and activities for teaching students about equivalent fractions and reducing fractions into simplest terms.

Compare and order pairs of fractions with these task cards, learning center activities, and worksheets.

These worksheets all feature fractions on number lines.

Download and print activities on calculating fractional parts of sets. (example: What is 3/4 of 24?)

Mixed number printables Includes basic mixed numbers, as well as adding and subtracting mixed numbers.

Print worksheets for learning about reciprocal fractions.

## 4: Fractions

The TT math line (a.k.a. our 4.0 version) is now a series of apps (one for each grade level)! This new format offers many advantages. Here's a quick list of the major benefits/enhancements.

• Our products are no longer dependent of the Flash Plug-In, which makes them easier to use.
• The 4.0 products were truly designed for phone and tablet use. Moreover, they can be used offline (for up to 6 lessons at a time without reconnecting to our servers). As a result, TT can now truly be used anywhere and at anytime (”on-the-go”, so to speak).
• This latest version has a host of new parent and student features such as a running course average (for parents), more and greater controls (for parents), and a place to work out the steps of a problem on-screen (for students). just to name a few.

## 4: Fractions

USE THIS HANDY TABLE FOR YOUR ANTENNA
MEASUREMENTS AND CONVERSIONS.
It also has many other uses!
Convert fractions to decimals and millimeters and reverse.
(1 INCH = 25.4 MM EXACTLY)
Read from left to right - pick your fraction, decimal, or mm measurement.
Example = convert 1/64" to mm . Find 1/64 and read to the right under mm !
You will see 0.3969 under the mm column.
Another example = convert 0.125 decimal to inches . Look down the decimal column
until you find 0.125 , then follow that line to the left to find 1/8 inches
or look in the right column for mm!

 fraction decimal mm fraction decimal mm fraction decimal mm 1/64 0.0156 0.3969 1 1/64 1.0156 25.7969 2 1/64 2.0156 51.1969 1/32 0.0313 0.7938 1 1/32 1.0313 26.1938 2 1/32 2.0313 51.5938 3/64 0.0469 1.1906 1 3/64 1.0469 26.5906 2 3/64 2.0469 51.9906 1/16 0.0625 1.5875 1 1/16 1.0625 26.9875 2 1/16 2.0625 52.3875 5/64 0.0781 1.9844 1 5/64 1.0781 27.3844 2 5/64 2.0781 52.7844 3/32 0.0938 2.3813 1 3/32 1.0938 27.7813 2 3/32 2.0938 53.1813 7/64 0.1094 2.7781 1 7/64 1.1094 28.1781 2 7/64 2.1094 53.5781 1/8 0.1250 3.1750 1 1/8 1.1250 28.5750 2 1/8 2.1250 53.9750 9/64 0.1406 3.5719 1 9/64 1.1406 28.9719 2 9/64 2.1406 54.3719 5/32 0.1563 3.9688 1 5/32 1.1563 29.3688 2 5/32 2.1563 54.7688 11/64 0.1719 4.3656 1 11/64 1.1719 29.7656 2 11/64 2.1719 55.1656 3/16 0.1875 4.7625 1 3/16 1.1875 30.1625 2 3/16 2.1875 55.5625 13/64 0.2031 5.1594 1 13/64 1.2031 30.5594 2 13/64 2.2031 55.9594 7/32 0.2188 5.5563 1 7/32 1.2188 30.9563 2 7/32 2.2188 56.3563 15/64 0.2344 5.9531 1 15/64 1.2344 31.3531 2 15/64 2.2344 56.7531 1/4 0.2500 6.3500 1 1/4 1.2500 31.7500 2 1/4 2.2500 57.1500 17/64 0.2656 6.7469 1 17/64 1.2656 32.1469 2 17/64 2.2656 57.5469 9/32 0.2813 7.1438 1 9/32 1.2813 32.5438 2 9/32 2.2813 57.9438 19/64 0.2969 7.5406 1 19/64 1.2969 32.9406 2 19/64 2.2969 58.3406 5/16 0.3125 7.9375 1 5/16 1.3125 33.3375 2 5/16 2.3125 58.7375 21/64 0.3281 8.3344 1 21/64 1.3281 33.7344 2 21/64 2.3281 59.1344 11/32 0.3438 8.7313 1 11/32 1.3438 34.1313 2 11/32 2.3438 59.5313 23/64 0.3594 9.1281 1 23/64 1.3594 34.5281 2 23/64 2.3594 59.9281 3/8 0.3750 9.5250 1 3/8 1.3750 34.9250 2 3/8 2.3750 60.3250 25/64 0.3906 9.9219 1 25/64 1.3906 35.3219 2 25/64 2.3906 60.7219 13/32 0.4063 10.3188 1 13/32 1.4063 35.7188 2 13/32 2.4063 61.1188 27/64 0.4219 10.7156 1 27/64 1.4219 36.1156 2 27/64 2.4219 61.5156 7/16 0.4375 11.1125 1 7/16 1.4375 36.5125 2 7/16 2.4375 61.9125 29/64 0.4531 11.5094 1 29/64 1.4531 36.9094 2 29/64 2.4531 62.3094 15/32 0.4688 11.9063 1 15/32 1.4688 37.3063 2 15/32 2.4688 62.7063 31/64 0.4844 12.3031 1 31/64 1.4844 37.7031 2 31/64 2.4844 63.1031 1/2 0.5000 12.7000 1 1/2 1.5000 38.1000 2 1/2 2.5000 63.5000 33/64 0.5156 13.0969 1 33/64 1.5156 38.4969 2 33/64 2.5156 63.8969 17/32 0.5313 13.4938 1 17/32 1.5313 38.8938 2 17/32 2.5313 64.2938 35/64 0.5469 13.8906 1 35/64 1.5469 39.2906 2 35/64 2.5469 64.6906 9/16 0.5625 14.2875 1 9/16 1.5625 39.6875 2 9/16 2.5625 65.0875 37/64 0.5781 14.6844 1 37/64 1.5781 40.0844 2 37/64 2.5781 65.4844 19/32 0.5938 15.0813 1 19/32 1.5938 40.4813 2 19/32 2.5938 65.8813 39/64 0.6094 15.4781 1 39/64 1.6094 40.8781 2 39/64 2.6094 66.2781 5/8 0.6250 15.8750 1 5/8 1.6250 41.2750 2 5/8 2.6250 66.6750 41/64 0.6406 16.2719 1 41/64 1.6406 41.6719 2 41/64 2.6406 67.0719 21/32 0.6563 16.6688 1 21/32 1.6563 42.0688 2 21/32 2.6563 67.4688 43/64 0.6719 17.0656 1 43/64 1.6719 42.4656 2 43/64 2.6719 67.8656 11/16 0.6875 17.4625 1 11/16 1.6875 42.8625 2 11/16 2.6875 68.2625 45/64 0.7031 17.8594 1 45/64 1.7031 43.2594 2 45/64 2.7031 68.6594 23/32 0.7188 18.2563 1 23/32 1.7188 43.6563 2 23/32 2.7188 69.0563 47/64 0.7344 18.6531 1 47/64 1.7344 44.0531 2 47/64 2.7344 69.4531 3/4 0.7500 19.0500 1 3/4 1.7500 44.4500 2 3/4 2.7500 69.8500 49/64 0.7656 19.4469 1 49/64 1.7656 44.8469 2 49/64 2.7656 70.2469 25/32 0.7813 19.8438 1 25/32 1.7813 45.2438 2 25/32 2.7813 70.6438 51/64 0.7969 20.2406 1 51/64 1.7969 45.6406 2 51/64 2.7969 71.0406 13/16 0.8125 20.6375 1 13/16 1.8125 46.0375 2 13/16 2.8125 71.4375 53/64 0.8281 21.0344 1 53/64 1.8281 46.4344 2 53/64 2.8281 71.8344 27/32 0.8438 21.4313 1 27/32 1.8438 46.8313 2 27/32 2.8438 72.2313 55/64 0.8594 21.8281 1 55/64 1.8594 47.2281 2 55/64 2.8594 72.6281 7/8 0.8750 22.2250 1 7/8 1.8750 47.6250 2 7/8 2.8750 73.0250 57/64 0.8906 22.6219 1 57/64 1.8906 48.0219 2 57/64 2.8906 73.4219 29/32 0.9063 23.0188 1 29/32 1.9063 48.4188 2 29/32 2.9063 73.8188 59/64 0.9219 23.4156 1 59/64 1.9219 48.8156 2 59/64 2.9219 74.2156 15/16 0.9375 23.8125 1 15/16 1.9375 49.2125 2 15/16 2.9375 74.6125 61/64 0.9531 24.2094 1 61/64 1.9531 49.6094 2 61/64 2.9531 75.0094 31/32 0.9688 24.6063 1 31/32 1.9688 50.0063 2 31/32 2.9688 75.4063 63/64 0.9844 25.0031 1 63/64 1.9844 50.4031 2 63/64 2.9844 75.8031 1" 1.0000 25.4000 2" 2.0000 50.8000 3" 3.0000 76.2000

MORE USEFUL CONVERSIONS
To convert decimal fractions of an inch to fractions of an inch.

Take the decimal fraction of feet and divide by 0.08333 (1/12th) and this will give you inches and decimals of an inch.
For example - 6.37 feet. Take the 0.37 feet and divide by 0.0833 = 4.44 inches .
So for 6.37 ft. we can also say 6 ft. - 4.44 in.
Now take the 0.44 in. and round it down to 0.4 which is 4/10"
We can now say 6.37' approximatly equals 6' - 4 4/10"

For fractions of an inch other than tenth's, take the decimal remainder of inches and divide by:
0.125 for the number of eighth's
0.0625 for the number of sixteenth's
0.03125 for the number of thirty-second's
0.015625 for the number of sixty-fourth's, and so on.

For example - 4.382 inches. For eighth's, divide the remainder, 0.382 by 0.125 and the answer is 3.056 so the fraction is a bit over 3 eighth's, therefore the answer ends up as approx. 4 3/8"

## MathHelp.com

For instance, here's how you would find and use a form of 1 to reduce 4 /8 :

To be very clear, the point of finding the common factor (in this case, the 4 's) is to allow you to convert part of the fraction to 1 . Since 4 /4 = 1 , then what I did above was the following:

Warning: Note how I switched from a fraction with products (in the numerator and denominator):

. to a product of fractions:

This switch is okay as long as you're multiplying:

. but it is very much NOT if you're adding. For instance:

The left-hand side above, being a fraction containing addition, is equal to 5 /6 , while the right-hand side above, being an addition containing fractions, is equal to 1 1 /2 , so the two expressions are not at all the same value. Just remember: For fractions, multiplying is way easier than adding. Now, to get back to business.

In addition to the canceling method I used above (with the pink 1 's), you may also have seen either of the following "shorthands" for cancellation:

Any of these formats is fine. The last two are probably simplest for your handwritten homework the first one is easier for typesetting.

If you have a regular (scientific, business, etc.) calculator that can handle fractions, then you can enter the fraction and then hit the "equals" button to get the reduced fraction. If you have a graphing calculator with a fraction command, then you can enter the fraction as a division (because 4 /8 means "four divided by eight"), and then convert to fraction form. Check your manual.

If your calculator can't handle fractions, or if the denominator is too large for the calculator to handle, here's how you do the reduction by hand.

#### Reduce to simplest form.

I'll grab my calculator and some scrap paper, and factor the numerator (top number) and denominator (bottom number). A quick shorthand for getting the prime factorization of each of these numbers is demonstrated below, in the stacked division (by prime numbers) of 2940 :

To find the factorization, I just read off the prime factors from around the outside of the upside-down division. From the above, I can see that 2940 factors as 2×2×3×5×7×7 .

Next, I'll factor the denominator, being the number 3150 :

Now I can reduce the fraction by canceling off the common factors:

The next section reviews mixed numbers and improper (or "vulgar") fractions.

## 4: Fractions

Note that the whole number-integral part is: empty
The decimal part is: .25 = 25 /100
Full simple fraction breakdown: 25/100
= 5/20
= 1/4

Scroll down to customize the precision point enabling 0.25 to be broken down to a specific number of digits. The page also includes 2-3D graphical representations of 0.25 as a fraction, the different types of fractions, and what type of fraction 0.25 is when converted.

##### Level of Precision for 0.25 as a Fraction

The level of precision are the number of digits to round to. Select a lower precision point below to break decimal 0.25 down further in fraction form. The default precision point is 5.

If the last trailing digit is ř" you can use the "round half up" and "round half down" options to round that digit up or down when you change the precision point.

For example 0.875 with a precision point of 2 rounded half up = 88/100, rounded half down = 87/100.

##### Graph Representation of 0.25 as a Fraction

Pie chart representation of the fractional part of 0.25

2D Chart 3D Chart

##### Numerator & Denominator for 0.25 as a Fraction

0.25 = 0 25 /100
numerator/denominator = 25 /100

##### Is 25 /100 a Mixed, Whole Number or Proper fraction?

A mixed number is made up of a whole number (whole numbers have no fractional or decimal part) and a proper fraction part (a fraction where the numerator (the top number) is less than the denominator (the bottom number). In this case the whole number value is empty and the proper fraction value is 25/100.

##### Can all decimals be converted into a fraction?

Not all decimals can be converted into a fraction. There are 3 basic types which include:

Terminating decimals have a limited number of digits after the decimal point.

Example: 156.54 = 156 54 /100

Recurring decimals have one or more repeating numbers after the decimal point which continue on infinitely.

Example: 7688.3333 = 7688 3333 /10000 = 333 /1000 = 33 /100 = 1 /3 (rounded)

Irrational decimals go on forever and never form a repeating pattern. This type of decimal cannot be expressed as a fraction.

## The Harmonic Numbers

The first few values of H(n) are:

The second table is the same fractions for H(n) but without simplifying the sums.
The numerators give the total number of cycles in all permutations of length n+1 and the fractions give the ratio (probability) of a random permutation on n+1 letters having exactly two cycles!
What can we say about H(n) ? Is any Harmonic number ever exactly an integer for instance?
Subtracting one Harmonic number from another larger one H(n) &ndash H(k) gives us the sum of a consecutive set of the unit fractions 1/(k+1) + . + 1/n . Are any of these ever integers?
Unfortunately, the answers are no no finite sum of the series of unit fraction starting at 1 or at another unit fraction will ever sum to a whole number.

### The sum of all unit fractions

Let's look at the first term: 1/2 ,
The next 2 terms: 1/3 + 1/4 but 1/3 > 1/4 so
 1 + 1 > 1 + 1 = 1 3 4 4 4 2

Now look at
the next 4 terms:
 1 + 1 + 1 + 1 > 1 + 1 + 1 + 1 = 1 5 6 7 8 8 8 8 8 2

similarly the sum of
the next 8 terms will exceed 1/2 .
Since the Harmonic series sum goes on for ever, then we can always find another batch of 2 n terms whose sum adds more than 1/2 to the total,
so the total is always larger than any given number, it never settles down to a fixed value, it grows for ever or "diverges".
This result has been known since at least 1650 (Pietro Mengoli).
• 75.11 The Noninteger Property of Sums of Reciprocals of Successive Integers Duane W. Detemple The Mathematical Gazette, Vol. 75, No. 472 (Jun., 1991), pages 193-194
Here is a proof that no consectuive reciprocals sum to an integer, simpler than that of the original of G.Polyà and G.Szegö of 1976.

### The Overhanging books puzzle

How many books will it take before the overhang exceeds the length of one book?
Answer: H(4)/2 = 25/24 which is bigger than 1 so: