A welldefined group of objects'

1

• set

• A well-defined group of objects.

• Examples
• 2,5,6,9
• N all natural numbers
• R all real numbers
• A All college students who receive some kind
  of
• financial aid

2

• natural numbers

• They are the counting numbers.

• The collection of natural numbers is usually
  called N and represented as
• N 1,2,3,4,

3

• whole numbers

• The set of numbers that includes zero and all of
  the natural numbers.

• W 0,1,2,3,

4

• integers

• The set of numbers consisting of the whole
  numbers and their opposites.

• Z -3, -2, -1, 0, 1, 2, 3,

5

• rational number

• A number that can be expressed as the ratio of
  two integers.

• Examples 2/3 , -(4/5) , 25/7
• Notes
• Every integer is also a rational number. For
  example, 3 can be expressed as 3/1
• All rational numbers have a decimal expression.
• For example
• 2/5 0.4
• 5/3 1.666

6

• irrational number

• A number that cannot be expressed as a ratio of
  two integers.

• Examples
• (the square root of 2) 1.4142135
• (the square root of 3) 1.7320508
• p (the relationship between the circumference
  and the
• diameter of a circle) ? 3.1415926
• e (the base of natural logarithms) ?
  2.7182818
• Note No irrational number can be written as a
  fraction or an ending decimal. All irrational
  numbers have non-ending, non-repeating decimal
  expressions.

7

• real numbers

• The combined set of rational numbers and
  irrational numbers

• Examples 4, 5/8, -7.21, ?
• 4, 5/8 and -7.21 are real numbers because they
  are rational numbers.
• ? is a real number because it is an irrational
  number.

8

• numbers - tree diagram

• Real Numbers

Rational
Irrational
Integer
Non-Integer
(-)
()
(-)
()
()
(-)
(0)
Examples 7 is real, rational, integer and
positive -2/3 is real, rational and negative
is real, irrational and positive 0.38 is real,
rational, non-integer and positive
9

• even number

• A natural number that is divisible by 2.
• The general form of an even number is 2n, where n
  is any whole number.

• Examples
• 12, 78 and 100 are even numbers.

10

• odd numbers

• A whole number that is not divisible by 2. The
  general form of an odd number is 2n 1, where n
  is any whole number.

• Examples
• 3, 7, 13 and 27

11

• digit

• The ten symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, and
  9.

• Example
• The number 215 has
• three digits 2, 1, and 5.

12

• commutative property of addition

• a b b a

• Example
• 5 12 12 5

13

• additive identity

• a 0 a
• 0 a a
• Examples
• 3 0 3
• 0 3/5 3/5
• (-7) 0 (-7)

• The number zero is called the additive identity
  because the sum of zero and any number is that
  number.

14

• associative property of addition

• (a b) c a (b c)

• Example
• (2 5) 8 2 (5 8)
• 7 8 2 13
• 15 15

15

• zero

• The additive identity the number that gives n
  when added to another number n.

• Example
• 7 0 7

16

• additive inverse

• The additive inverse of any number x is the
  number that gives zero when added to x.

• (-a) is the additive inverse of a, since a (-a)
  0
• Examples
• (-4) is the additive inverse of 4
• 7 is the additive inverse of -7

17

• difference

• The result of subtracting two numbers.

• 7 - 4 3
• 3 is the difference
• (-7) (-2) (-5)
• (-5) is the difference

18

• factor and product

• Factor one of two or more expressions that are
  multiplied together to get a product.

• Examples
• 3 x 5 15
• 3 and 5 are factors
• 15 is the product
• In the expression 2ab, 2 , a and b are factors.

19

• commutative property of multiplication

• a ? b b ? a

• Example
• (-7) x (4) (4) x (-7)
• (-28) (-28)

20

• multiplicative identity

• The number 1 is the multiplicative identity
  because multiplying 1 times any number gives that
  number.

• Examples
• 8 ? 1 8
• 1 ? (-5) -5
• (3/7) ? 1 (3/7)

21

• zero property of multiplication

• The product of zero and any number is zero.

• Examples
• 3 ? 0 0
• 0 ? 3.5 0
• (-12) ? 0 0
• 0 ? (2/3) 0

22

• associative property of multiplication

• (a ? b) ? c a ? (b ? c)

• Example
• (7 x 4) x 3 7 x (4 x 3)
• 28 x 3 7 x 12
• 84 84

23

• division by zero is not defined

• The result of dividing any number by zero is not
  defined. That operation does not have a result
  and therefore it is invalid.

• Examples
• 2/0 not defined (undefined)
• 25 ? 0 not defined

24

• distributive property

• a(b c) ab ac

• Examples
• 8(4 5) (8 ? 4) (8 ? 5)
• 8(9) (32) (40)
• 72 72
• 2(x5) 2x 10

25

• signed multiplication and division

• Multiplying
• ( ) ( ) ( )
• ( - ) ( - ) ( )
• ( ) ( - ) ( - )
• ( - ) ( ) ( - )

• Dividing
• ( ) ( ) ( )
• ( - ) ( - ) ( )
• ( ) ( - ) ( - )
• ( - ) ( ) ( - )

26

• multiplicative inverse

• Multiplicative inverse or simply inverse of a
  number is another number such that the product of
  both equals 1.
• Examples
• 1/3 is the inverse of 3, because their product is
  1.
• a/b is the inverse of b/a because their product
  is 1

27

• dividend

• In a b c,
• a is the dividend.

• In 14 7 2,
• 14 is the dividend.

28

• divisor

• In 15 3 5,
• 3 is the divisor.

• In a b c, b is the divisor.

29

• terms in a division

• dividend (divisor ? quotient) remainder

• Division 17 3 ?
• There is no whole number that multiplied by 3
  that will give 17. We use 5 as a quotient and
  get a remainder of 2
• 17 (3 ? 5) 2
• 17 dividend
• 3 divisor
• 5 quotient
• 2 remainder

30

• exact division

• In the operation a b c, a is the dividend,
  b is the divisor and the result c is the
  quotient.

• Example
• 15 3 5
• 15 is the dividend, 3 is the divisor and 5 is the
  quotient, because 3 ? 5 15
• Note that zero divided by any number is equal to
  zero (e.g. 0 3 0).

31

• removing parenthesis

• Examples
• 5 (7 x) 5 7 - x
• 5 (7 x) 5 - 7 x
• a (-3 b) a 3 b
• a (-3 b) a 3 b

• When a parenthesis is preceded by a sign , it
  is removed without changing all of the sign(s)
  inside.
• When a parenthesis is preceded by a sign , it
  is removed by changing all of the signs inside.

32

• symbols of inclusion

• Examples
• a3 a (b2)
• a3 a b 2
• a3 a b 2
• a1 a b a a2 ab

• Some expressions in arithmetic and algebra use
  operations included in other operations.
• Their symbols, by hierarchical order (from
  inside to outside) are
• Parenthesis
• Brackets
• Braces
• ()

33

• order of operations

• Perform the operations inside the symbol of
  inclusion (parenthesis, brackets and/or braces)
  and above and below each fraction bar. Start with
  the innermost inclusion symbol.
• Perform all multiplications and divisions in the
  order they appear from left to right.
• Perform all additions and subtractions in the
  order they appear from left to right.

• Example
• 23 x (15) ? 4 3 6 Work inside the
  parenthesis
• 8 x 6 ? 4 3 6 Evaluate exponents
  
• 48 ? 4 3 6 Multiply
• 12 3 6 Add and subtract
  
• 9

34

• multiple

• A multiple of a number is the product of that
  number and any other whole number. Zero is a
  multiple of every number.

• 20 is a multiple of 4 since 20 4 x 5
• 86 is a multiple of 7 since 86 7 x 12

35

• prime number

• A number whose only factors are itself and 1.

• Examples
• 2, 3, 5, 7, 11, 13, 17, 19,
• 23, 29, 31, 37, 41, 43,
• 47, 53,

36

• composite number

• A natural number that is not prime.

• Examples
• 6 2 ? 3
• 50 2 ? 5 ? 5
• 6 and 50 are composite numbers they are not
  prime.
• They have factors.

37

• common factor

• Examples
• 15 x 5
• 12 x 2 x 2
• 3 is a common factor of 15 and 12.

• A factor of two or more numbers.

3
3
38

• greatest common factor (GCF)

• Examples
• 12 3 x 2 x 2
• 20 5 x 2 x 2
• 4 is the GCF of 12 and 20.
• 3 x 3 x 5 x 2 90
• 7 x 3 x 5 x 2 210
• 2 x 2 x 3 x 5 60
• 30 is the GCF of 90, 210 and 60.

• The largest number that divides two or more
  numbers evenly.

39

• least common multiple (LCM)

• The smallest non-zero number that is a multiple
  of two or more numbers.

• Example
• 3 3 x 1
• 10 2 x 5
• 4 2 x 2
• The LCM of 3, 10 and 4 is 60
• LCM 3 x 1 x 2 x 2 x 5 60

40

• numerator

• The top part of a fraction.

• Examples
• In 4/5, the numerator is 4.
• In (2x)/7, the numerator is 2x.
• In (ab)/3, the numerator is ab.

41

• denominator

• The bottom part of a fraction.

• Examples
• In 4/5, the denominator is 5.
• In A/B, the denominator is B.

42

• proper fraction

• A fraction whose numerator is less than its
  denominator.

• Examples
• 2/5 0.4 and 2 ? 5
• 12/37 0.324 and 12 ? 37
• 1/100 0.01 and 1 ? 100
• 234/823 ? 0.2843256 and 234 ? 823
• Note that the decimal expression of a proper
  function is not greater than 1.

43

• improper fraction

• A fraction with a numerator that is greater than
  the denominator.

• Examples
• 3/2, 25/7 and 5/4
• Note that all of these numbers are greater than
  1
• 3/2 1.5 and 3 ? 2
• 25/7 ? 3.5714285 and 25 ? 7
• 5/4 1.25 and 5 ?4

44

• like fractions

• Fractions that have the same denominators.

• Examples
• 2/5, 7/5, 3/5 are like fractions.
• -1/10, 5/10 and 9/10 are like fractions.

45

• mixed number

• A number written as a whole number and a
  fraction.
• Note Every mixed number is equivalent to an
  improper fraction. Also all types of fractions
  (proper, improper and mixed numbers) are
  considered non-integers.

• Example
• This number is read as three and two fifths.
  It is equivalent to the fraction .

46

• equivalent fractions

• Fractions that reduce to the same number.

• Examples
• and are equivalent fractions.
• and are equivalent fractions.

47

• lowest terms (LT)

• Simplest form when the GCF of the numerator and
  the denominator of a fraction is 1

• Examples
• 35/100 in LT is 7/20
• 12/27 in LT is 4/9
• So 30/105 in LT or simplest form is 2/7

48

• simplifying

• Examples
• Simplifying 24/56 reduces the fraction to 3/7
• Simplifying (4x2) /(12xy) reduces the fraction to
  (x/3y)

• Reducing to lowest terms

49

• reciprocal

• The number which, when multiplied times a
  particular fraction, gives a result of 1.

• Examples
• 1/7 is reciprocal of 7 since
• (7) x (1/7) 1.
• (Note that 7 is the reciprocal of 1/7.)

50

• opposite

• The number which, when added to a particular
  number, gives a result of 0.

• Examples
• -7 is the reciprocal of 7 since
• (7) (-7) 0.
• (Note that 7 is the reciprocal of -7.)

51

• multiplicative inverse

• The reciprocal of a number is the multiplicative
  inverse of the number

• Examples
• 1/7 is the multiplicative inverse of 7
• 7 is the multiplicative inverse of 1/7
• -5 is the multiplicative inverse of (-1/5)
• (-1/5) is the multiplicative inverse of (-5)

52

• least common denominator (LCD)

• The smallest multiple of the denominators of two
  or more fractions.

• Examples
• 2/3 , 3/5 LCD 15 (3 x 5)
• 3/5, 7/10, 9/20 LCD 20 (5 x 2 x 2)
• 1/7, 5/28, 7/12 LCD 86 (7 x 2 x 2 x 3)

53

• cross product

• A product found by multiplying the numerator of
  one fraction by the denominator of another
  fraction and the denominator of the first
  fraction by the numerator of the second fraction.

• Examples
• a/5 3/4 cross product ax4 5x3
• 3/a 2/5 cross product 3x5 ax2
• 5/7 a/4 cross product 5x4 7xa
• 2/9 1/a cross product 2xa 9x1

54

• decimal number

• The numbers in the base 10 number system, having
  one or more places to the right of a decimal
  point.

• Examples
• 2.1
• -3.15
• 0.004
• 321.9056

55

• terminating decimal

• Examples
• 12/100 0 .12
• 47/5 9.4
• 3/8 0.375

• A fraction whose decimal representation contains
  a finite number of digits.

56

• non-terminating repeating decimal

• Examples
• 5/6 0.83333
• 14/99 0.14141414

• A fraction whose decimal representation contains
  a block of digits that repeats indefinitely.

57

• non-repeating decimals

• These numbers have infinite non-repeating digits
  in their decimal places.
• These numbers are decimal expressions of
  irrational numbers.

• Examples
• 1.414213562373095.
• 1.73205080756887
• ? 3.14159265358979

58

• percent

• A fraction, or ratio, in which the denominator
  is assumed to be 100. The symbol is used for
  percent. A percent can be written as a fraction
  and as a decimal expression

• Examples
• 3 3/100
• 3 0.03
• 33 33/100 0.33
• 125 125/100 1.25

59

• positive number

• A real number greater than zero.

• Examples
• 2, 4.5 , 7/12, 0.0003

60

• negative number

• A real number that is less than zero.

• Examples
• -3
• -5/7
• -0.54
• -237.35

61

• absolute value

• The distance of a number from zero in other
  words, the positive value of a number.

• Example
• 3 3
• -3 3
• Notice that 3 and (3) are at the same distance
  from zero on the number line

62

• inequality

• A mathematical expression which shows that two
  quantities are not equal.

• Examples
• 15 lt 23 (15 is less than 23)
• -3 gt - 8 (-3 is greater than -8)
• 0.05 lt 0.5 (0.05 is less than 0.5)
• x gt a (x is greater than a)

63

• constant and variable

• Constant a value that does not change
• Variable a value that changes

• Examples
• 3 is a constant
• X is a variable
• In the expression 5x p, the constants are 5
  and p , while x is the variable.

64

• variable

• Example
• In the expression 2x -7y,
• x and y are variables.

• A letter used to represent a number value in an
  expression or an equation.

65

• evaluate

• To substitute number values into an expression.

• Example
• Evaluate the expression
• x2y
• For x3 and y(-1)
• x 2y (3) 2(-1)
• 3 (-2) 3 2 1

66

• formula

• An equation that states a rule or a fact.

• Examples
• (a b)2 a2 2ab b2 (algebra)
• Where a and b can either be constants or
  variables
• A w x h (geometry)
• Where A is the area of a rectangle whose width
  is w and whose height is h

67

• base, exponent and power

• Exponent a number that indicates the operation
  of repeated multiplication for the same factor.
• Base the factor
• Power the product of n equal factors. It is
  denoted as
• a x a x a x a an , where n is the
  exponent.

• Examples
• 3 x 3 32 9 , the base is 3, the exponent
  is 2 and the power is the resulting 9
• Other examples
• (-1) ? (-1) ? (-1) (-1)3 -1
• 2 ? 2 ? 2 ? 2 24 16
• (-2) ? (-2) ? (-2) ? (-2) (-2)4 16
• (-2) ? (-2) ? (-2) (-2)3 -8

68

• equation

• A mathematical statement that says that two
  expressions have the same value it is a number
  sentence with an equal sign, . It may also
  contain a variable(s).

• Examples
• x 8 15
• 3x 2 y
• 1

69

• root

• The root of an equation is the same as the
  solution to the equation.

• Examples
• For the equation x - 58, x13 is a root
  because when plugged into the equation it makes
  the equality true.
• The equation x2 x 2 0 has two roots, x(-2)
  and x 1.

70

• inverse operations

• Two operations that have the opposite effect,
  such as addition and subtraction.

• Examples
• (7 3) 3 7
• (8 4) x 4 8
• (11 5) 5 11
• (5 x 3) 3 5

71

• equivalent equations

• Two equations whose solutions are the same.

• Example
• 2x 7 5 and 4x 14 10 are equations that
  are equivalent.

72

• ratio

• Examples
• 2/5 A day care center has 2 teachers for every
  5 toddlers.
• 15/1000 In a certain region there are 15
  physician per 1000 inhabitants.

• A pair of numbers that compares different types
  of units.

73

• proportion

• An equation of fractions in the form a/b c/d

• Examples
• 4/7 32/56
• 20/4 100/20
• Note A proportion may include one or more
  variables.
• For example
• 5/x 7/21
• x/y 4/13

74

• rate

• Examples
• Velocity 60 miles per hour
• Sales 230 cars per month
• Population density 40 inhabitants per square mile

• A ratio that compares different kinds of units.

75

• square root

• The square root of x ( ) is the number
  that, when multiplied by itself, gives the number
  x.

• Example
• 5 is a square root of 25 since 5 x 5 25
• (-5) is another square root of 25 since (-5)(-5)
  25

76

• pi

• The relationship between the circumference of a
  circle and its diameter.

• Pi is represented by the Greek letter p.
• Its decimal expression is p 3.141695.
• Note Since p is an irrational number, its
  decimal expression in non-ending and
  non-repeating.

77

• counting principle

• If a first event has n outcomes and a second
  event has m outcomes, then the first event
  followed by the second event has n times m
  outcomes.

• Example
• If I have to choose among 3 different blouses
  and 5 different skirts, then there will be 15
  distinct outfits that I can wear (3x5 15).

78

• permutation

• A way to arrange things in which order is
  important.

• Example
• In how many ways can you select tow objects from
  a collection of 4 objects if the order is
  important?
• Collection a, b, c, d,
• Possible orderings
• ab, ac, ad, ba, bc, bd, ca, cb, cd, da, db, dc
• Note that there are 12 permutations (how many
  ways would you obtain if the order does not
  matter?)

79

• combination

• A selection in which order is not important.

• Example
• In how many ways can you select two objects from
  a collection of four distinct object if the order
  is not important? Collection A,B,C,D
• Possible selections AB, AC, AD, BC, BD, CD
• Six ways!
• Note that since you considered AB, you do not
  consider BA as an option.

80

• logic

• Logic is the study of sound reasoning.