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7.6: Chapter 5 Review Exercises


Chapter Review Exercises

Add and Subtract Polynomials

Determine the Degree of Polynomials

In the following exercises, determine the type of polynomial.

1. (16x^2−40x−25)

2. (5m+9)

Answer

binomial

3. (−15)

4. (y^2+6y^3+9y^4)

Answer

other polynomial

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

5. (4p+11p)

6. (−8y^3−5y^3)

Answer

(−13y^3)

7. ((4a^2+9a−11)+(6a^2−5a+10))

8. ((8m^2+12m−5)−(2m^2−7m−1))

Answer

(6m^2+19m−4)

9. ((y^2−3y+12)+(5y^2−9))

10. ((5u^2+8u)−(4u−7))

Answer

(5u^2+4u+7)

11. Find the sum of (8q^3−27) and (q^2+6q−2).

12. Find the difference of (x^2+6x+8) and (x^2−8x+15).

Answer

(2x^2−2x+23)

In the following exercises, simplify.

13. (17mn^2−(−9mn^2)+3mn^2)

14. (18a−7b−21a)

Answer

(−7b−3a)

15. (2pq^2−5p−3q^2)

16. ((6a^2+7)+(2a^2−5a−9))

Answer

(8a^2−5a−2)

17. ((3p^2−4p−9)+(5p^2+14))

18. ((7m^2−2m−5)−(4m^2+m−8))

Answer

(−3m+3)

19. ((7b^2−4b+3)−(8b^2−5b−7))

20. Subtract ((8y^2−y+9)) from ( (11y^2−9y−5) )

Answer

(3y^2−8y−14)

21. Find the difference of ((z^2−4z−12)) and ((3z^2+2z−11))

22. ((x^3−x^2y)−(4xy^2−y^3)+(3x^2y−xy^2))

Answer

(x^3+2x^2y−4xy^2)

23. ((x^3−2x^2y)−(xy^2−3y^3)−(x^2y−4xy^2))

Evaluate a Polynomial Function for a Given Value of the Variable

In the following exercises, find the function values for each polynomial function.

24. For the function (f(x)=7x^2−3x+5) find:
a. (f(5)) b. (f(−2)) c. (f(0))

Answer

a. 165 b. 39 c. 5

25. For the function (g(x)=15−16x^2), find:
a. (g(−1)) b. (g(0)) c. (g(2))

26. A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial function (h(t)=−16t^2+640) gives the height of the glasses t seconds after they were dropped. Find the height of the glasses when (t=6).

Answer

The height is 64 feet.

27. A manufacturer of the latest soccer shoes has found that the revenue received from selling the shoes at a cost of (p) dollars each is given by the polynomial (R(p)=−5p^2+360p). Find the revenue received when (p=110) dollars.

Add and Subtract Polynomial Functions

In the following exercises, find a. ((f + g)(x)) b. ((f + g)(3)) c. ((f − g)(x) d. ((f − g)(−2))

28. (f(x)=2x^2−4x−7) and (g(x)=2x^2−x+5)

Answer

a. ((f+g)(x)=4x^2−5x−2)
b. ((f+g)(3)=19)
c. ((f−g)(x)=−3x−12)
d. ((f−g)(−2)=−6)

29. (f(x)=4x^3−3x^2+x−1) and (g(x)=8x^3−1)

Properties of Exponents and Scientific Notation

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

30. (p^3·p^{10})

Answer

(p^{13})

31. (2·2^6)

32. (a·a^2·a^3)

Answer

(a^6)

33. (x·x^8)

34. (y^a·y^b)

Answer

(y^{a+b})

35. (dfrac{2^8}{2^2})

36. (dfrac{a^6}{a})

Answer

(a^5)

37. (dfrac{n^3}{n^{12}})

38. (dfrac{1}{x^5})

Answer

(dfrac{1}{x^4})

39. (3^0)

41. ((14t)^0)

42. (12a^0−15b^0)

Answer

(−3)

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

43. (6^{−2})

44. ((−10)^{−3})

Answer

(−dfrac{1}{1000})

45. (5·2^{−4})

46. ((8n)^{−1})

Answer

(dfrac{1}{8n})

47. (y^{−5})

48. (10^{−3})

Answer

(dfrac{1}{1000})

49. (dfrac{1}{a^{−4}})

50. (dfrac{1}{6^{−2}})

Answer

(36)

51. (−5^{−3})

52. ( left(−dfrac{1}{5} ight)^{−3})

Answer

(−dfrac{1}{25})

53. (−(12)^{−3})

54. ((−5)^{−3})

Answer

(−dfrac{1}{125})

55. (left(dfrac{5}{9} ight)^{−2})

56. (left(−dfrac{3}{x} ight)^{−3})

Answer

(dfrac{x^3}{27})

In the following exercises, simplify each expression using the Product Property.

57. ((y^4)^3)

58. ((3^2)^5)

Answer

(3^{10})

59. ((a^{10})^y)

60. (x^{−3}·x^9)

Answer

(x^5)

61. (r^{−5}·r^{−4})

62. ((uv^{−3})(u^{−4}v^{−2}))

Answer

(dfrac{1}{u^3v^5})

63. ((m^5)^{−1})

64. (p^5·p^{−2}·p^{−4})

Answer

(dfrac{1}{m^5})

In the following exercises, simplify each expression using the Power Property.

65. ((k−2)^{−3})

66. (dfrac{q^4}{q^{20}})

Answer

(dfrac{1}{q^{16}})

67. (dfrac{b^8}{b^{−2}})

68. (dfrac{n^{−3}}{n^{−5}})

Answer

(n^2)

In the following exercises, simplify each expression using the Product to a Power Property.

69. ((−5ab)^3)

70. ((−4pq)^0)

Answer

(1)

71. ((−6x^3)^{−2})

72. ((3y^{−4})^2)

Answer

(dfrac{9}{y^8})

In the following exercises, simplify each expression using the Quotient to a Power Property.

73. (left(dfrac{3}{5x} ight)^{−2})

74. (left(dfrac{3xy^2}{z} ight)^4)

Answer

(dfrac{81x^4y^8}{z^4})

75. ((4p−3q^2)^2)

In the following exercises, simplify each expression by applying several properties.

76. ((x^2y)^2(3xy^5)^3)

Answer

(27x^7y^{17})

77. ((−3a^{−2})^4(2a^4)^2(−6a^2)^3)

78. (left(dfrac{3xy^3}{4x^4y^{−2}} ight)^2left(dfrac{6xy^4}{8x^3y^{−2}} ight)^{−1})

Answer

(dfrac{3y^4}{4x^4})

In the following exercises, write each number in scientific notation.

79. (2.568)

80. (5,300,000)

Answer

(5.3×10^6)

81. (0.00814)

In the following exercises, convert each number to decimal form.

82. (2.9×10^4)

Answer

(29,000)

83. (3.75×10^{−1})

84. (9.413×10^{−5})

Answer

(0.00009413)

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

85. ((3×10^7)(2×10^{−4}))

86. ((1.5×10^{−3})(4.8×10^{−1}))

Answer

(0.00072)

87. (dfrac{6×10^9}{2×10^{−1}})

88. (dfrac{9×10^{−3}}{1×10^{−6}})

Answer

(9,000)

Multiply Polynomials

Multiply Monomials

In the following exercises, multiply the monomials.

89. ((−6p^4)(9p))

90. (left(frac{1}{3}c^2 ight)(30c^8))

Answer

(10c^{10})

91. ((8x^2y^5)(7xy^6))

92. ( left(frac{2}{3}m^3n^6 ight)left(frac{1}{6}m^4n^4 ight))

Answer

(dfrac{m^7n^{10}}{9})

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

93. (7(10−x))

94. (a^2(a^2−9a−36))

Answer

(a^4−9a^3−36a^2)

95. (−5y(125y^3−1))

96. ((4n−5)(2n^3))

Answer

(8n^4−10n^3)

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using:

a. the Distributive Property b. the FOIL method c. the Vertical Method.

97. ((a+5)(a+2))

98. ((y−4)(y+12))

Answer

(y^2+8y−48)

99. ((3x+1)(2x−7))

100. ((6p−11)(3p−10))

Answer

(18p^2−93p+110)

In the following exercises, multiply the binomials. Use any method.

101. ((n+8)(n+1))

102. ((k+6)(k−9))

Answer

(k^2−3k−54)

103. ((5u−3)(u+8))

104. ((2y−9)(5y−7))

Answer

(10y^2−59y+63)

105. ((p+4)(p+7))

106. ((x−8)(x+9))

Answer

(x^2+x−72)

107. ((3c+1)(9c−4))

108. ((10a−1)(3a−3))

Answer

(30a^2−33a+3)

Multiply a Polynomial by a Polynomial

In the following exercises, multiply using a. the Vertical Method.

109. ((x+1)(x^2−3x−21))

110. ((5b−2)(3b^2+b−9))

Answer

(15b^3−b^2−47b+18)

In the following exercises, multiply. Use either method.

111. ((m+6)(m^2−7m−30))

112. ((4y−1)(6y^2−12y+5))

Answer

(24y^2−54y^2+32y−5)

Multiply Special Products

In the following exercises, square each binomial using the Binomial Squares Pattern.

113. ((2x−y)^2)

114. ((x+dfrac{3}{4})^2)

Answer

(x^2+dfrac{3}{2}x+dfrac{9}{16})

115. ((8p^3−3)^2)

116. ((5p+7q)^2)

Answer

(25p^2+70pq+49q^2)

In the following exercises, multiply each pair of conjugates using the Product of Conjugates.

117. ((3y+5)(3y−5))

118. ((6x+y)(6x−y))

Answer

(36x^2−y^2)

119. ((a+dfrac{2}3b)(a−dfrac{2}{3}b))

120. ((12x^3−7y^2)(12x^3+7y^2))

Answer

(144x^6−49y^4)

121. ((13a^2−8b4)(13a^2+8b^4))

Divide Monomials

Divide Monomials

In the following exercises, divide the monomials.

122. (72p^{12}÷8p^3)

Answer

(9p^9)

123. (−26a^8÷(2a^2))

124. (dfrac{45y^6}{−15y^{10}})

Answer

(−3y^4)

125. (dfrac{−30x^8}{−36x^9})

126. (dfrac{28a^9b}{7a^4b^3})

Answer

(dfrac{4a^5}{b^2})

127. (dfrac{11u^6v^3}{55u^2v^8})

128. (dfrac{(5m^9n^3)(8m^3n^2)}{(10mn^4)(m^2n^5)})

Answer

(dfrac{4m^9}{n^4})

129. (dfrac{(42r^2s^4)(54rs^2)}{(6rs^3)(9s)})

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial

130. ((54y^4−24y^3)÷(−6y^2))

Answer

(−9y^2+4y)

131. (dfrac{63x^3y^2−99x^2y^3−45x^4y^3}{9x^2y^2})

132. (dfrac{12x^2+4x−3}{−4x})

Answer

(−3x−1+dfrac{3}{4x})

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

133. ((4x^2−21x−18)÷(x−6))

134. ((y^2+2y+18)÷(y+5))

Answer

(y−3+dfrac{33}{q+6})

135. ((n^3−2n^2−6n+27)÷(n+3))

136. ((a^3−1)÷(a+1))

Answer

(a^2+a+1)

Divide Polynomials using Synthetic Division

In the following exercises, use synthetic Division to find the quotient and remainder.

137. (x^3−3x^2−4x+12) is divided by (x+2)

138. (2x^3−11x^2+11x+12) is divided by (x−3)

Answer

(2x^2−5x−4;space0)

139. (x^4+x^2+6x−10) is divided by (x+2)

Divide Polynomial Functions

In the following exercises, divide.

140. For functions (f(x)=x^2−15x+45) and (g(x)=x−9), find a. (left(dfrac{f}{g} ight)(x))
b. (left(dfrac{f}{g} ight)(−2))

Answer

a. (left(dfrac{f}{g} ight)(x)=x−6)
b. (left(dfrac{f}{g} ight)(−2)=−8)

141. For functions (f(x)=x^3+x^2−7x+2) and (g(x)=x−2), find a. (left(dfrac{f}{g} ight)(3))

Use the Remainder and Factor Theorem

In the following exercises, use the Remainder Theorem to find the remainder.

142. (f(x)=x^3−4x−9) is divided by (x+2)

Answer

(−9)

143. (f(x)=2x^3−6x−24) divided by (x−3)

In the following exercises, use the Factor Theorem to determine if (x−c) is a factor of the polynomial function.

144. Determine whether (x−2) is a factor of (x^3−7x^2+7x−6)

Answer

no

145. Determine whether (x−3) is a factor of (x^3−7x^2+11x+3)

Chapter Practice Test

1. For the polynomial (8y^4−3y^2+1)

a. Is it a monomial, binomial, or trinomial? b. What is its degree?

Answer

a. trinomial b. 4

2. ((5a^2+2a−12)(9a^2+8a−4))

3. ((10x^2−3x+5)−(4x^2−6))

Answer

(6x^2−3x+11)

4. (left(−dfrac{3}{4} ight)^3)

5. (x^{−3}x^4)

Answer

(x)

6. (5^65^8)

7. ((47a^{18}b^{23}c^5)^0)

Answer

(1)

8. (4^{−1})

9. ((2y)^{−3})

Answer

(dfrac{1}{8y^3})

10. (p^{−3}·p^{−8})

11. (dfrac{x^4}{x^{−5}})

Answer

(x^9)

12. ((3x^{−3})^2)

13. (dfrac{24r^3s}{6r^2s^7})

Answer

(dfrac{4r}{s^6})

14. ((x4y9x−3)2)

15. ((8xy^3)(−6x^4y^6))

Answer

(−48x^5y^9)

16. (4u(u^2−9u+1))

17. ((m+3)(7m−2))

Answer

(21m^2−19m−6)

18. ((n−8)(n^2−4n+11))

19. ((4x−3)^2)

Answer

(16x^2−24x+9)

20. ((5x+2y)(5x−2y))

21. ((15xy^3−35x^2y)÷5xy)

Answer

(3y^2−7x )

22. ((3x^3−10x^2+7x+10)÷(3x+2))

23. Use the Factor Theorem to determine if (x+3) a factor of (x^3+8x^2+21x+18).

Answer

yes

24. a. Convert 112,000 to scientific notation.
b. Convert (5.25×10^{−4}) to decimal form.

In the following exercises, simplify and write your answer in decimal form.

25. ((2.4×10^8)(2×10^{−5}))

Answer

(4.4×10^3)

26. (dfrac{9×10^4}{3×10^{−1}})

27. For the function (f(x)=6x^2−3x−9) find:
a. (f(3)) b. (36) b. (21) c. (-9)

28. For (f(x)=2x^2−3x−5) and (g(x)=3x^2−4x+1), find
a. ((f+g)(x)) b. ((f+g)(1))
c. ((f−g)(x)) d. ((f−g)(−2))

29. For functions (f(x)=3x^2−23x−36) and (g(x)=x−9), find
a. (left(dfrac{f}{g} ight)(x)) b. (left(dfrac{f}{g} ight)(3))

Answer

a. (left(dfrac{f}{g} ight)(x)=3x+4)
b. (left(dfrac{f}{g} ight)(3)=13)

30. A hiker drops a pebble from a bridge (240) feet above a canyon. The function (h(t)=−16t^2+240) gives the height of the pebble (t) seconds after it was dropped. Find the height when (t=3).


Chapter 7, Problem Exercises 5

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Exploring the U.S. Census

Exploring the U.S. Census gives social science students and researchers alike the tools to understand, extract, process, and analyze data from the decennial census, the American Community Survey, and other data collected by the U.S. Census Bureau. Donnelly's text provides a thorough background on the data collection methods, structures, and potential pitfalls of the census for unfamiliar researchers, collecting information previously available only in widely disparate sources into one handy guide. Hands-on, applied exercises at the end of the chapters help readers dive into the data.

Along the way, the author shows how best to analyze census data with open-source software and tools. Readers can freely evaluate the data on their own computers, in keeping with the free and open data provided by the Census Bureau. By placing the census in the context of the open data movement, this text makes the history and practice of the census relevant so readers can understand what a crucial resource the census is for research and knowledge.


1.1 Concurrency everywhere

1.3 Two classic distributed computing problems

Chapter 2. Two-Process Systems

2.1 Elementary graph theory

2.5 Two-process task solvability

Chapter 3. Elements of Combinatorial Topology

3.3 Standard constructions

3.7 Simplicial and continuous approximations

Chapter 4. Colorless Wait-Free Computation

4.3 The computational power of wait-free colorless immediate snapshots

Chapter 5. Solvability of Colorless Tasks in Different Models

5.2 t-Resilient layered snapshot protocols

5.3 Layered snapshots with k-set agreement

5.5 Message-passing protocols

Chapter 6. Byzantine-Resilient Colorless Computation

6.2 Byzantine communication abstractions

6.3 Byzantine set agreement

6.4 Byzantine barycentric agreement

6.5 Byzantine task solvability

6.6 Byzantine shared memory

Chapter 7. Simulations and Reductions

Chapter 8. Read-Write Protocols for General Tasks

Chapter 9. Manifold Protocols

9.2 Layered immediate snapshot protocols

9.3 No set agreement from manifold protocols

9.4 Set agreement vs. weak symmetry breaking

10.1 Consensus and path connectivity

10.2 Immediate snapshot model and connectivity

10.3 k-Set agreement and -connectivity

10.4 Immediate snapshot model and k-connectivity

Chapter 11. Wait-Free Computability for General Tasks

11.1 Inherently colored tasks: the hourglass task

11.2 Solvability for colored tasks

11.3 Algorithm implies map

11.4 Map implies algorithm

11.5 A sufficient topological condition

Chapter 12. Renaming and Oriented Manifolds

12.1 An upper bound: renaming with names

12.2 Weak symmetry breaking

12.5 A lower bound for -renaming

Chapter 13. Task Solvability in Different Models

13.4 Carrier maps and shellable complexes

Chapter 14. Simulations and Reductions for Colored Tasks

14.4 Layered snapshot from read-write

14.5 Immediate snapshot from snapshot

14.6 Immediate snapshot from layered immediate snapshot

14.7 Snapshot from layered snapshot

Chapter 15. Classifying Loop Agreement Tasks

15.1 The fundamental group

Chapter 16. Immediate Snapshot Subdivisions

16.1 A glimpse of discrete geometry


The solutions for selected exercises from each chapter can be found below. Be careful about looking at the solutions too quickly make sure you’ve given yourself time to wrestle with the concepts you just learned before looking at a solution. Also, there are several ways to solve many of the exercises, and the solutions only show one possible way to complete each exercise.

I haven’t included solutions for Chapters 12-14 and 18-20, because the exercises for those chapters are really projects in themselves. If you’re having trouble with an exercise from one of those chapters consider posting on Stack Overflow, r/learnpython, or get in touch.


Chapter 5 Canonical Correlation Analysis (CCA)

Suppose we observe a random sample of (n) bivariate observations [ mathbf z_1=(x_1,y_1)^ op , ldots , mathbf z_n=(x_n,y_n)^ op. ] If we are interested in exploring possible dependence between the (x_i) ’s and (y_i) ’s then among the first things we would do would be to obtain a scatterplot of the (x_i) ’s against the (y_i) ’s and calculate the correlation coefficient. Recall that the sample correlation coefficient is defined by [egin r=operatorname>(x,y)&=frac<>>>sqrt<>>> &=frac^n (x_i-ar)(y_i-ar)>^n (x_i-ar)^2 ight)^ <1/2>left(sum_^n (y_i-ar)^2 ight)^<1/2>> ag <5.1>end] where (ar=n^<-1>sum_^n x_i) and (ar=n^<-1>sum_^n y_i) are the sample means.

Recall that the sample correlation is a scale-free measure of the strength of the linear dependence between the (x_i) ’s and the (y_i) ’s.

In this chapter we investigate the multivariate analogue of this question. Instead of our bivariate observations being a pair of scalars, suppose instead that we are given two different random vectors (mathbf x) and (mathbf y) . In otherwords, for each subject/case (i) we have observations (_^n.)

Multivariate data structures can be understood better if we look at low-dimensional projections of the data. The question is, given a sample (_^) , what is a sensible way to assess and describe the strength of the linear dependence between the two vectors?

Canonical correlation analysis (CCA) gives an answer to this question in terms of the best low-dimensional linear projections of the (mathbf x) and (mathbf y) random variables. In a comparable way to PCA, ‘best’ in CCA is defined in terms of maximizing correlations. A key role is played by the singular value decomposition (SVD) introduced in Chapter 3.


5.6 The Gestalt Principles of Perception

In the early part of the 20th century, Max Wertheimer published a paper demonstrating that individuals perceived motion in rapidly flickering static images—an insight that came to him as he used a child’s toy tachistoscope. Wertheimer, and his assistants Wolfgang Köhler and Kurt Koffka, who later became his partners, believed that perception involved more than simply combining sensory stimuli. This belief led to a new movement within the field of psychology known as Gestalt psychology. The word gestalt literally means form or pattern, but its use reflects the idea that the whole is different from the sum of its parts. In other words, the brain creates a perception that is more than simply the sum of available sensory inputs, and it does so in predictable ways. Gestalt psychologists translated these predictable ways into principles by which we organize sensory information. As a result, Gestalt psychology has been extremely influential in the area of sensation and perception (Rock & Palmer, 1990).

Gestalt perspectives in psychology represent investigations into ambiguous stimuli to determine where and how these ambiguities are being resolved by the brain. They are also aimed at understanding sensory and perception as processing information as groups or wholes instead of constructed wholes from many small parts. This perspective has been supported by modern cognitive science through fMRI research demonstrating that some parts of the brain, specifically the lateral occipital lobe, and the fusiform gyrus, are involved in the processing of whole objects, as opposed to the primary occipital areas that process individual elements of stimuli (Kubilius, Wagemans & Op de Beeck, 2011).

One Gestalt principle is the figure-ground relationship. According to this principle, we tend to segment our visual world into figure and ground. Figure is the object or person that is the focus of the visual field, while the ground is the background. As the figure below shows, our perception can vary tremendously, depending on what is perceived as figure and what is perceived as ground. Presumably, our ability to interpret sensory information depends on what we label as figure and what we label as ground in any particular case, although this assumption has been called into question (Peterson & Gibson, 1994 Vecera & O’Reilly, 1998).

The concept of figure-ground relationship explains why this image can be perceived either as a vase or as a pair of faces.

Another Gestalt principle for organizing sensory stimuli into meaningful perception is proximity. This principle asserts that things that are close to one another tend to be grouped together, as the figure below illustrates.

The Gestalt principle of proximity suggests that you see (a) one block of dots on the left side and (b) three columns on the right side.

How we read something provides another illustration of the proximity concept. For example, we read this sentence like this, notl iket hiso rt hat. We group the letters of a given word together because there are no spaces between the letters, and we perceive words because there are spaces between each word. Here are some more examples: Cany oum akes enseo ft hiss entence? What doth es e wor dsmea n?

We might also use the principle of similarity to group things in our visual fields. According to this principle, things that are alike tend to be grouped together (figure below). For example, when watching a football game, we tend to group individuals based on the colors of their uniforms. When watching an offensive drive, we can get a sense of the two teams simply by grouping along this dimension.

When looking at this array of dots, we likely perceive alternating rows of colors. We are grouping these dots according to the principle of similarity.

Two additional Gestalt principles are the law of continuity (or good continuation) and closure. The law of continuity suggests that we are more likely to perceive continuous, smooth flowing lines rather than jagged, broken lines (figure below). The principle of closure states that we organize our perceptions into complete objects rather than as a series of parts (figure below).

Good continuation would suggest that we are more likely to perceive this as two overlapping lines, rather than four lines meeting in the center.

Closure suggests that we will perceive a complete circle and rectangle rather than a series of segments.

According to Gestalt theorists, pattern perception, or our ability to discriminate among different figures and shapes, occurs by following the principles described above. You probably feel fairly certain that your perception accurately matches the real world, but this is not always the case. Our perceptions are based on perceptual hypotheses: educated guesses that we make while interpreting sensory information. These hypotheses are informed by a number of factors, including our personalities, experiences, and expectations. We use these hypotheses to generate our perceptual set. For instance, research has demonstrated that those who are given verbal priming produce a biased interpretation of complex ambiguous figures (Goolkasian & Woodbury, 2010).

Template Approach

Ulrich Neisser (1967), author of one of the first cognitive psychology textbook suggested pattern recognition would be simplified, although abilities would still exist, if all the patterns we experienced were identical. According to this theory, it would be easier for us to recognize something if it matched exactly with what we had perceived before. Obviously the real environment is infinitely dynamic producing countless combinations of orientation, size. So how is it that we can still read a letter g whether it is capitalized, non-capitalized or in someone else hand writing? Neisser suggested that categorization of information is performed by way of the brain creating mental templates, stored models of all possible categorizable patterns (Radvansky & Ashcraft, 2014). When a computer reads your debt card information it is comparing the information you enter to a template of what the number should look like (has a specific amount of numbers, no letters or symbols…). The template view perception is able to easily explain how we recognize pieces of our environment, but it is not able to explain why we are still able to recognize things when it is not viewed from the same angle, distance, or in the same context.

In order to address the shortfalls of the template model of perception, the feature detection approach to visual perception suggests we recognize specific features of what we are looking at, for example the straight lines in an H versus the curved line of a letter C. Rather than matching an entire template-like pattern for the capital letter H, we identify the elemental features that are present in the H. Several people have suggested theories of feature-based pattern recognition, one of which was described by Selfridge (1959) and is known as the pandemonium model suggesting that information being perceived is processed through various stages by what Selfridge described as mental demons, who shout out loud as they attempt to identify patterns in the stimuli. These pattern demons are at the lowest level of perception so after they are able to identify patterns, computational demons further analyze features to match to templates such as straight or curved lines. Finally at the highest level of discrimination, cognitive demons which allow stimuli to be categorized in terms of context and other higher order classifications, and the decisions demon decides among all the demons shouting about what the stimuli is which while be selected for interpretation.

Selfridge’s pandemonium model showing the various levels of demons which make estimations and pass the information on to the next level before the decision demon makes the best estimation to what the stimuli is. Adapted from Lindsay and Norman (1972).

Although Selfridges ideas regarding layers of shouting demons that make up our ability to discriminate features of our environment, the model actually incorporates several ideas that are important for pattern recognition. First, at its foundation, this model is a feature detection model that incorporates higher levels of processing as the information is processed in time. Second, the Selfridge model of many different shouting demons incorporates ideas of parallel processing suggesting many different forms of stimuli can be analyzed and processed to some extent at the same time. Third and finally, the model suggests that perception in a very real sense is a series of problem solving procedures where we are able to take bits of information and piece it all together to create something we are able to recognize and classify as something meaningful.

In addition to sounding initially improbable by being based on a series of shouting fictional demons, one of the main critiques of Selfridge’s demon model of feature detection is that it is primarily a bottom-up, or data-driven processing system. This means the feature detection and processing for discrimination all comes from what we get out of the environment. Modern progress in cognitive science has argued against strictly bottom-up processing models suggesting that context plays an extremely important role in determining what you are perceiving and discriminating between stimuli. To build off previous models, cognitive scientist suggested an additional top-down, or conceptually-driven account in which context and higher level knowledge such as context something tends to occur in or a persons expectations influence lower-level processes.

Finally the most modern theories that attempt to describe how information is processed for our perception and discrimination are known as connectionist models. Connectionist models incorporate an enormous amount of mathematical computations which work in parallel and across series of interrelated web like structures using top-down and bottom-up processes to narrow down what the most probably solution for the discrimination would be. Each unit in a connectionist layer is massively connected in a giant web with many or al the units in the next layer of discrimination. Within these models, even if there is not many features present in the stimulus, the number of computations in a single run for discrimination become incredibly large because of all the connections that exist between each unit and layer.

The Depths of Perception: Bias, Prejudice, and Cultural Factors

In this chapter, you have learned that perception is a complex process. Built from sensations, but influenced by our own experiences, biases, prejudices, and cultures , perceptions can be very different from person to person. Research suggests that implicit racial prejudice and stereotypes affect perception. For instance, several studies have demonstrated that non-Black participants identify weapons faster and are more likely to identify non-weapons as weapons when the image of the weapon is paired with the image of a Black person (Payne, 2001 Payne, Shimizu, & Jacoby, 2005). Furthermore, White individuals’ decisions to shoot an armed target in a video game is made more quickly when the target is Black (Correll, Park, Judd, & Wittenbrink, 2002 Correll, Urland, & Ito, 2006). This research is important, considering the number of very high-profile cases in the last few decades in which young Blacks were killed by people who claimed to believe that the unarmed individuals were armed and/or represented some threat to their personal safety.

SUMMARY

Gestalt theorists have been incredibly influential in the areas of sensation and perception. Gestalt principles such as figure-ground relationship, grouping by proximity or similarity, the law of good continuation, and closure are all used to help explain how we organize sensory information. Our perceptions are not infallible, and they can be influenced by bias, prejudice, and other factors.

Openstax Psychology text by Kathryn Dumper, William Jenkins, Arlene Lacombe, Marilyn Lovett and Marion Perlmutter licensed under CC BY v4.0. https://openstax.org/details/books/psychology

Exercises

Review Questions:

1. According to the principle of ________, objects that occur close to one another tend to be grouped together.

2. Our tendency to perceive things as complete objects rather than as a series of parts is known as the principle of ________.

3. According to the law of ________, we are more likely to perceive smoothly flowing lines rather than choppy or jagged lines.

4. The main point of focus in a visual display is known as the ________.

Critical Thinking Question:

1. The central tenet of Gestalt psychology is that the whole is different from the sum of its parts. What does this mean in the context of perception?

2. Take a look at the following figure. How might you influence whether people see a duck or a rabbit?

Personal Application Question:

1. Have you ever listened to a song on the radio and sung along only to find out later that you have been singing the wrong lyrics? Once you found the correct lyrics, did your perception of the song change?

Key Takeaways

Review Questions:

Critical Thinking Question:

1. This means that perception cannot be understood completely simply by combining the parts. Rather, the relationship that exists among those parts (which would be established according to the principles described in this chapter) is important in organizing and interpreting sensory information into a perceptual set.

2. Playing on their expectations could be used to influence what they were most likely to see. For instance, telling a story about Peter Rabbit and then presenting this image would bias perception along rabbit lines.

closure: organizing our perceptions into complete objects rather than as a series of parts

figure-ground relationship: segmenting our visual world into figure and ground

Gestalt psychology: field of psychology based on the idea that the whole is different from the sum of its parts

good continuation: (also, continuity) we are more likely to perceive continuous, smooth flowing lines rather than jagged, broken lines

pattern perception: ability to discriminate among different figures and shapes

perceptual hypothesis: educated guess used to interpret sensory information

principle of closure: organize perceptions into complete objects rather than as a series of parts

proximity: things that are close to one another tend to be grouped together

similarity: things that are alike tend to be grouped together

Review Questions

According to the principle of ________, objects that occur close to one another tend to be grouped together.

Our tendency to perceive things as complete objects rather than as a series of parts is known as the principle of ________.

According to the law of ________, we are more likely to perceive smoothly flowing lines rather than choppy or jagged lines.

The main point of focus in a visual display is known as the ________.

Critical Thinking Question

The central tenet of Gestalt psychology is that the whole is different from the sum of its parts. What does this mean in the context of perception?

Take a look at the following figure. How might you influence whether people see a duck or a rabbit?

Answer: Playing on their expectations could be used to influence what they were most likely to see. For instance, telling a story about Peter Rabbit and then presenting this image would bias perception along rabbit lines.

Personal Application Question

Have you ever listened to a song on the radio and sung along only to find out later that you have been singing the wrong lyrics? Once you found the correct lyrics, did your perception of the song change?

Glossary


5.1 Sensation versus Perception

What does it mean to sense something? Sensory receptors are specialized neurons that respond to specific types of stimuli. When sensory information is detected by a sensory receptor, sensation has occurred. For example, light that enters the eye causes chemical changes in cells that line the back of the eye. These cells relay messages, in the form of action potentials (as you learned when studying biopsychology), to the central nervous system. The conversion from sensory stimulus energy to action potential is known as transduction. Transduction represents the first step toward perception and is a translation process where different types of cells react to stimuli creating a signal processed by the central nervous system resulting in what we experience as a sensations. Sensations allow organisms to sense a face, and smell smoke when there is a fire.

Perceptions on the other hand, require organizing and understanding the incoming sensation information. In order for sensations to be useful, we must first add meaning to those sensations, which create our perceptions of those sensations. Sensations allow us to see a red burner, but perceptions entail the understanding and representation of the characteristic hot. Also, a sensation would be hearing a loud, shrill tone, whereas a perception would be the classification and understanding of that sounds as a fire alarm. Throughout this chapter sensations and perceptions will be discussed as separate events, whereas in reality, sensations and perceptions can be more accurately thought of as occurring along a continued where boundaries are more fluent between where a sensation ends and a perception begins.

You have probably known since elementary school that we have five senses: vision, hearing (audition), smell (olfaction), taste (gustation), and touch (somatosensation). It turns out that this notion of five senses is extremely oversimplified. We also have sensory systems that provide information about balance (the vestibular sense), body position and movement (proprioception and kinesthesia), pain (nociception), and temperature (thermoception), and each one of these sensory systems has different receptors tuned to transduce different stimuli. The vision system absorbs light using rod and cone receptors located at the back of the eyes, sound is translated via tiny hair like receptors known as cilia inside the inner ear, smell and taste work together most of the time to absorb chemicals found in airborne particles and food via chemically sensitive cilia in the nasal cavity and clusters of chemical receptors on the tongue. Touch is particularly interesting because it is made up of responses from many different types of receptors found within the skin that send signals to the central nervous system in response to temperature, pressure, vibration, and disruption of the skin such as stretching and tearing.

Free nerve endings embedded in the skin that allow humans to perceive the various differences in our immediate environment. Adapted from Pinel, 2009.

The sensitivity of a given sensory system to the relevant stimuli can be expressed as an absolute threshold. Absolute threshold refers to the minimum amount of stimulus energy that must be present for the stimulus to be detected 50% of the time. Another way to think about this is by asking how dim can a light be or how soft can a sound be and still be detected half of the time. The sensitivity of our sensory receptors can be quite amazing. It has been estimated that on a clear night, the most sensitive sensory cells in the back of the eye can detect a candle flame 30 miles away (Okawa & Sampath, 2007). Under quiet conditions, the hair cells (the receptor cells of the inner ear) can detect the tick of a clock 20 feet away (Galanter, 1962). Additionally, one teaspoon of sugar can be tasted within two gallons of water, and the human olfactory system can detect the scent of one drop of perfume throughout a six room apartment.

It is also possible for us to get messages that are presented below the threshold for conscious awareness—these are called subliminal messages. A stimulus reaches a physiological threshold when it is strong enough to excite sensory receptors and send nerve impulses to the brain: This is an absolute threshold. A message below that threshold is said to be subliminal: The message is processed, but we are not consciously aware of it. Over the years, there has been a great deal of speculation about the use of subliminal messages in advertising, rock music, and self-help audio programs to influence consumer behavior. Research has demonstrated in laboratory settings, people can process and respond to information outside of awareness. But this does not mean that we obey these messages like zombies in fact, hidden messages have little effect on behavior outside the laboratory (Kunst-Wilson & Zajonc, 1980 Rensink, 2004 Nelson, 2008 Radel, Sarrazin, Legrain, & Gobancé, 2009 Loersch, Durso, & Petty, 2013). Studies attempting to influence movie goers to purchase more popcorn, and reduced smoking habits demonstrated little to no success further suggesting subliminal messages are mostly ineffective in producing specific behavior (Karremans, Stroebe & Claus, 2006). However, neuroimaging studies have demonstrated clear neural activity related to the processing of subliminal stimuli stimuli (Koudier & Dehaene, 2007). Additionally, Krosnick, Betz, Jussim & Lynn (1992) found that participants who were presented images of dead bodies or buckets of snakes for several milliseconds (subliminal priming), were more likely to rate a neutral image of a woman with a neutral facial expression as more unlikable compared to participants who were shown more pleasant images (kittens and bridal couples). This demonstrates that although we may not be aware of the stimuli presented to us, we are processing it on a neural level, and also that although subliminal priming usually is not strong enough to force unwanted purchases, it may influence our perceptions of things we encounter in the environment following the subliminal priming.

Absolute thresholds are generally measured under incredibly controlled conditions in situations that are optimal for sensitivity. Sometimes, we are more interested in how much difference in stimuli is required to detect a difference between them. This is known as the just noticeable difference (JND, mentioned briefly in the above study comparing color perceptions of Chinese and Dutch participants) or difference threshold. Unlike the absolute threshold, the difference threshold changes depending on the stimulus intensity. As an example, imagine yourself in a very dark movie theater. If an audience member were to receive a text message on her cell phone which caused her screen to light up, chances are that many people would notice the change in illumination in the theater. However, if the same thing happened in a brightly lit arena during a basketball game, very few people would notice. The cell phone brightness does not change, but its ability to be detected as a change in illumination varies dramatically between the two contexts. Ernst Weber proposed this theory of change in difference threshold in the 1830s, and it has become known as Weber’s law.

Webers Law: Each of the various senses has its own constant ratios determining difference thresholds.

Webers ideas about difference thresholds influenced concepts of signal detection theory which state that our abilities to detect a stimulus depends on sensory factors (like the intensity of the stimulus, or the presences of other stimuli being processed) as well as our psychological state (you are sleepy because you stayed up studying the previous night). Human factors engineers who design control consoles for planes and cars use signal detection theory all the time in order to asses situations pilots or drivers may experience such as difficulty in seeing and interpreting controls on extremely bright days.

PERCEPTION

Although are perceptions are built from sensations, not all sensations result in perception.”

While our sensory receptors are constantly collecting information from the environment, it is ultimately how we interpret that information that affects how we interact with the world. Perception refers to the way sensory information is organized, interpreted, and consciously experienced. Perception involves both bottom-up and top-down processing. Bottom-up processing refers to the fact that perceptions are built from sensory input, stimuli from the environment. On the other hand, how we interpret those sensations is influenced by our available knowledge, our experiences, and our thoughts related to the stimuli we are experiencing. This is called top-down processing.

One way to think of this concept is that sensation is a physical process, whereas perception is psychological. For example, upon walking into a kitchen and smelling the scent of baking cinnamon rolls, the sensation is the scent receptors detecting the odor of cinnamon, but the perception may be “Mmm, this smells like the bread Grandma used to bake when the family gathered for holidays.” Sensation is a signal from any of our six senses. Perception is the brain’s response to these signals. When we see our professor speaking in the front of the room, we sense the visual and auditory signals coming from them and we perceive that they are giving a lecture about our psychology class.

Although our perceptions are built from sensations, not all sensations result in perception. In fact, we often don’t perceive stimuli that remain relatively constant over prolonged periods of time. This is known as sensory adaptation. Imagine entering a classroom with an old analog clock. Upon first entering the room, you can hear the ticking of the clock as you begin to engage in conversation with classmates or listen to your professor greet the class, you are no longer aware of the ticking. The clock is still ticking, and that information is still affecting sensory receptors of the auditory system. The fact that you no longer perceive the sound demonstrates sensory adaptation and shows that while closely associated, sensation and perception are different. Additionally, when you walk into a dark movie theater after being outside on a bright day you will notice it is initially extremely difficult to see. After a couple minutes you experience what is known as dark adaptation which tends to take about 8 minutes for cones (visual acuity and color), and about 30 minutes for the cones in your retina to adapt (light, dark, depth and distance) (Hecht & Mendelbaum, 1938 Klaver, Wolfs, Vingerling, Hoffman, & de Jong, 1998). If you are wondering why it takes so long to adapt to darkness, in order to change the sensitivity of rods and cones, they must first undergo a complex chemical change associated with protein molecules which does not happen immediately. Now that you have adapted to the darkens of the theater, you have survived marathon watching the entire Lord of the Rings series, and you are emerging from the theater a seemly short ten hours after entering the theater, you may experience the process of light adaptation, barring it is still light outside. During light adaptation, the pupils constrict to reduce the amount of light flooding onto the retina and sensitivity to light is reduced for both rods and cones which takes usually less than 10 minutes (Ludel, 1978). So why is the process of raising sensitivity to light to adapt to darkness more complex than lowering sensitivity to adapt to light? Caruso (2007) has suggested that a more gradual process is involved in darkness adaptation due to humans tendency over the course of evolution to slowly adjust to darkness as the sun sets over the horizon.

There is another factor that affects sensation and perception: attention. Attention plays a significant role in determining what is sensed versus what is perceived. Imagine you are at a party full of music, chatter, and laughter. You get involved in an interesting conversation with a friend, and you tune out all the background noise. If someone interrupted you to ask what song had just finished playing, you would probably be unable to answer that question.

One of the most interesting demonstrations of how important attention is in determining our perception of the environment occurred in a famous study conducted by Daniel Simons and Christopher Chabris (1999). In this study, participants watched a video of people dressed in black and white passing basketballs. Participants were asked to count the number of times the team in white passed the ball. During the video, a person dressed in a black gorilla costume walks among the two teams. You would think that someone would notice the gorilla, right? Nearly half of the people who watched the video didn’t notice the gorilla at all, despite the fact that he was clearly visible for nine seconds. Because participants were so focused on the number of times the white team was passing the ball, they completely tuned out other visual information. Failure to notice something that is completely visible because of a lack of attention is called inattentional blindness. More recent work evaluated inattention blindness related to cellphone use. Hyman, Boss, Wise, McKenzie & Caggiano (2010) classified participants based on whether they were walking while talking on their cell phone, listening to an MP3 player, walking without any electronics or walking as a pair. Participants were not aware that while they walked through the square a unicycling clown would ride right in front of them. After the students reached the outside of the square they were stopped and asked if they noticed the unicycling clown that rode in front of them. Cell phone users were found to walk more slowly, change directions more often, pay less attention to others around them and were also the most frequent group to report they did not noticed the unicycling clown. David Strayer and Frank Drews additionally examined cell phone use in a series of driving simulators and found that even when participants looked directly at the objects in the driving environment, they were less likely to create a durable memory of those objects if they were talking on a cell phone. This pattern was obtained for objects of both high and low relevance for their driving safety suggesting little meaningful cognitive analysis of objects in the driving environment outside the restricted focus of attention while maintaining a cell phone conversation. Additionally, in-vehicle conversations did not interfere with driving as much as cell phone conversations as Strayer and Drews suggest, drivers are better able to synchronize the processing demands of driving with in-vehicle conversations compared to cell-phone conversations. Overall it is apparent that directing the focus of our attention can lead to sometimes serious impairments of other information, and it appears cell phones can have a particularly dramatic impact on information processing while performing other tasks.

In a similar experiment to the activity above, researchers tested inattentional blindness by asking participants to observe images moving across a computer screen. They were instructed to focus on either white or black objects, disregarding the other color. When a red cross passed across the screen, about one third of subjects did not notice it (figure below) (Most, Simons, Scholl, & Chabris, 2000).

Nearly one third of participants in a study did not notice that a red cross passed on the screen because their attention was focused on the black or white figures. (credit: Cory Zanker)

Motivation can also affect perception. Have you ever been expecting a really important phone call and, while taking a shower, you think you hear the phone ringing, only to discover that it is not? If so, then you have experienced how motivation to detect a meaningful stimulus can shift our ability to discriminate between a true sensory stimulus and background noise. This motivational aspect of expectation in conversation additionally may be why such strong inattentional blindness has been found in relation to cell phone use. The ability to identify a stimulus when it is embedded in a distracting background is called signal detection theory.

Signal detection theory: A theory explaining explaining how various factors influence our ability to detect weak signals in our environment.

Signal detection theory also explains why a mother is awakened by a quiet murmur from her baby but not by other sounds that occur while she is asleep. This also applies to air traffic controller communication, pilot and driver control panels as discussed previously, and even the monitoring of patient vital information while a surgeon performs surgery. In the case of air traffic controllers, the controllers need to be able to detect planes among many signals (blips) that appear on the radar screen and follow those planes as they move through the sky. In fact, the original work of the researcher who developed signal detection theory was focused on improving the sensitivity of air traffic controllers to plane blips (Swets, 1964).

Our perceptions can also be affected by our beliefs, values, prejudices, expectations, and life experiences. As you will see later in this chapter, individuals who are deprived of the experience of binocular vision during critical periods of development have trouble perceiving depth (Fawcett, Wang, & Birch, 2005). The shared experiences of people within a given cultural context can have pronounced effects on perception. For example, Marshall Segall, Donald Campbell, and Melville Herskovits (1963) published the results of a multinational study in which they demonstrated that individuals from Western cultures were more prone to experience certain types of visual illusions than individuals from non-Western cultures, and vice versa. One such illusion that Westerners were more likely to experience was the Müller-Lyer illusion (figure below): The lines appear to be different lengths, but they are actually the same length.

In the Müller-Lyer illusion, lines appear to be different lengths although they are identical. (a) Arrows at the ends of lines may make the line on the right appear longer, although the lines are the same length. (b) When applied to a three-dimensional image, the line on the right again may appear longer although both black lines are the same length.

These perceptual differences were consistent with differences in the types of environmental features experienced on a regular basis by people in a given cultural context. People in Western cultures, for example, have a perceptual context of buildings with straight lines, what Segall’s study called a carpentered world (Segall et al., 1966). In contrast, people from certain non-Western cultures with an uncarpentered view, such as the Zulu of South Africa, whose villages are made up of round huts arranged in circles, are less susceptible to this illusion (Segall et al., 1999). It is not just vision that is affected by cultural factors. Indeed, research has demonstrated that the ability to identify an odor, and rate its pleasantness and its intensity, varies cross-culturally (Ayabe-Kanamura, Saito, Distel, Martínez-Gómez, & Hudson, 1998). In terms of color vision across cultures, research has found derived color terms for brown, orange and pink hues do appear to be influenced by cultural differences (Zollinger, 1988).

Children described as thrill seekers are more likely to show taste preferences for intense sour flavors (Liem, Westerbeek, Wolterink, Kok, & de Graaf, 2004), which suggests that basic aspects of personality might affect perception. Furthermore, individuals who hold positive attitudes toward reduced-fat foods are more likely to rate foods labeled as reduced fat as tasting better than people who have less positive attitudes about these products (Aaron, Mela, & Evans, 1994).

SUMMARY

Sensation occurs when sensory receptors detect sensory stimuli. Perception involves the organization, interpretation, and conscious experience of those sensations. All sensory systems have both absolute and difference thresholds, which refer to the minimum amount of stimulus energy or the minimum amount of difference in stimulus energy required to be detected about 50% of the time, respectively. Sensory adaptation, selective attention, and signal detection theory can help explain what is perceived and what is not. In addition, our perceptions are affected by a number of factors, including beliefs, values, prejudices, culture, and life experiences.

Openstax Psychology text by Kathryn Dumper, William Jenkins, Arlene Lacombe, Marilyn Lovett and Marion Perlmutter licensed under CC BY v4.0. https://openstax.org/details/books/psychology

Exercises

Review Questions:

1. ________ refers to the minimum amount of stimulus energy required to be detected 50% of the time.

c. just noticeable difference

2. Decreased sensitivity to an unchanging stimulus is known as ________.

d. inattentional blindness

3. ________ involves the conversion of sensory stimulus energy into neural impulses.

b. inattentional blindness

4. ________ occurs when sensory information is organized, interpreted, and consciously experienced.

Critical Thinking Question:

1. Not everything that is sensed is perceived. Do you think there could ever be a case where something could be perceived without being sensed?

2. Please generate a novel example of how just noticeable difference can change as a function of stimulus intensity.

Personal Application Question :

1. Think about a time when you failed to notice something around you because your attention was focused elsewhere. If someone pointed it out, were you surprised that you hadn’t noticed it right away?

just noticeable difference

Answers to Exercises

Review Questions:

Critical Thinking Question:

1. This would be a good time for students to think about claims of extrasensory perception. Another interesting topic would be the phantom limb phenomenon experienced by amputees.

2. There are many potential examples. One example involves the detection of weight differences. If two people are holding standard envelopes and one contains a quarter while the other is empty, the difference in weight between the two is easy to detect. However, if those envelopes are placed inside two textbooks of equal weight, the ability to discriminate which is heavier is much more difficult.

absolute threshold: minimum amount of stimulus energy that must be present for the stimulus to be detected 50% of the time

bottom-up processing: system in which perceptions are built from sensory input

inattentional blindness: failure to notice something that is completely visible because of a lack of attention

just noticeable difference: difference in stimuli required to detect a difference between the stimuli

perception: way that sensory information is interpreted and consciously experienced

sensation: what happens when sensory information is detected by a sensory receptor

sensory adaptation: not perceiving stimuli that remain relatively constant over prolonged periods of time

signal detection theory: change in stimulus detection as a function of current mental state

subliminal message: message presented below the threshold of conscious awareness

top-down processing: interpretation of sensations is influenced by available knowledge, experiences, and thoughts

transduction: conversion from sensory stimulus energy to action potential


Geometry: The Line and the Circle

Geometry: The Line and the Circle is an undergraduate text with a strong narrative that is written at the appropriate level of rigor for an upper-level survey or axiomatic course in geometry. Starting with Euclid's Elements , the book connects topics in Euclidean and non-Euclidean geometry in an intentional and meaningful way, with historical context.

The line and the circle are the principal characters driving the narrative. In every geometry considered&mdashwhich include spherical, hyperbolic, and taxicab, as well as finite affine and projective geometries&mdashthese two objects are analyzed and highlighted. Along the way, the reader contemplates fundamental questions such as: What is a straight line? What does parallel mean? What is distance? What is area?

There is a strong focus on axiomatic structures throughout the text. While Euclid is a constant inspiration and the Elements is repeatedly revisited with substantial coverage of Books I, II, III, IV, and VI, non-Euclidean geometries are introduced very early to give the reader perspective on questions of axiomatics. Rounding out the thorough coverage of axiomatics are concluding chapters on transformations and constructibility. The book is compulsively readable with great attention paid to the historical narrative and hundreds of attractive problems.

The authors have provided a supplemental book of laboratory projects offering guided explorations to accompany topics found in the book. These projects use GeoGebra, a free interactive application. Download the GeoGebra Labs and a zip file of GeoGebra Lab starter files.

Readership

Undergraduate students interested in geometry.

Reviews & Endorsements

A fun and masterful road to learning what geometry is actually about. This is likely an ideal text for use in training secondary level teachers who will teach this glorious subject. From constructions to proofs of results to the larger meaning of these results within the overarching context of Geometry, this text is there to guide students and assist them in constructing their own mastery of the subject. It is a beautiful text with real depth and detail and will be of great value to anyone who wishes to know "Just what is geometry about, after all is said and done?". Highly recommended!


Connecting your certificate to your account

CMS Analysis with CRAB requires that the user's authentication credential is mapped to the a globally unique username. Currently authentication is based on grid certificate, where user is identified by the so called DN and we use the CERN primary computing account as username. If you use a certificated from CERN, this operation is fully transparent and you need to do no nothing but be aware of what you CERN username is. If you are using a grid certificate issued by a Certification Authority other than CERN CA, then read and follow the instructions in the Username for CRAB page to make sure your certificate is correctly mapped to your account.


Watch the video: WOD: Test obratnosti (November 2021).