The document provides an overview of topics related to numbers and quantities that may appear on the ACT exam, including: I. Real and complex number systems, rational vs irrational numbers, integers, whole numbers, natural numbers. Converting between fractions, mixed numbers, and improper fractions. Comparing fractions. (Paragraphs 1-13) II. Counting consecutive integers, using the number line to locate numbers and find distances. Radicals, simplifying square roots. (Paragraphs 14-23) III. The complex number system, imaginary numbers, and the complex plane. Greatest common factors and least common multiples, using calculators to find them. (Paragraphs 24-25)

1. Table of Contents
1
I. Number & Quantity
II. Algebra
III. Functions
IV. Geometry
V. Statistics & Probability
VI. Integrating Essential Skills

2. I. Number & Quantity
2
A. Real Number System
B. Complex Number System
C. Vectors
D. Matrices

3. Number Map
3

4. Number Map
Real Numbers
4
 A real number is a number that has a location on the
number line.
 On the ACT, imaginary numbers are numbers that involve
the square root of a negative number.

5. Number Map
Rational Numbers vs. Irrational Numbers
5
A rational number is a number that can be expressed
as a ratio of two integers.
Irrational numbers are real numbers – they have
locations on the number line – but they can’t be
expressed precisely as a fraction or decimal.
Examples:
Rational Numbers:
Irrational Numbers:

6. Number Map
Integers
6
Integers are whole numbers and their opposites
They include zero as well
… –3, –2, –1, 0, 1, 2, 3, …

7. Number Map
Whole Numbers
7
All the traditional counting numbers and zero
No negative numbers
0, 1, 2, 3, …

8. Number Map
Natural Numbers
8
Only the traditional counting numbers
No negative numbers or zero
1, 2, 3, …

9. 1. Properties of Numbers:
Parity (Even/Odd) & Sign (Positive/Negative)
 To predict whether a sum, difference, or product will be
even or odd, positive or negative just take simple numbers
like 1 and 2 and see what happens.
Example
The product of an even number and an
odd number is…
A. always even
B. always odd
C. always negative
D. always positive
E. sometimes odd
9

10. #1
10
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

11. #1
10
Answer: D
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

12. 1. Properties of Numbers
Converting Improper Fractions to Mixed
Numbers
11
Mixed Number  Improper Fraction
1. Divide the
denominator into
the numerator to
get a whole
number quotient
with a remainder
2. The quotient
becomes the
whole number part
of the mixed
number
3. The remainder
becomes the new
numerator with the
same denominator

13. 1. Properties of Numbers
Converting Improper Fractions to Mixed
Numbers
11
Mixed Number  Improper Fraction
1. Divide the
denominator into
the numerator to
get a whole
number quotient
with a remainder
2. The quotient
becomes the
whole number part
of the mixed
number
3. The remainder
becomes the new
numerator with the
same denominator
11/4 = 2 Rem. 3

14. 1. Properties of Numbers
Converting Mixed Numbers to Improper
Fractions
 Improper fractions: fractions where the numerator is larger
than the denominator
 Mixed numbers: numbers with a whole number portion and a
fractional portion
12

15. 1. Properties of Numbers
Converting Mixed Numbers to Improper
Fractions
 Improper fractions: fractions where the numerator is larger
than the denominator
 Mixed numbers: numbers with a whole number portion and a
fractional portion
12
Mixed Number  Improper Fraction
1. Multiply the
whole number part
by the
denominator
2. Then add the
product to the
numerator
3. The result is the
new numerator
(over the same
denominator)

16. 1. Properties of Numbers
Converting Mixed Numbers to Improper
Fractions
 Improper fractions: fractions where the numerator is larger
than the denominator
 Mixed numbers: numbers with a whole number portion and a
fractional portion
12
Mixed Number  Improper Fraction
1. Multiply the
whole number part
by the
denominator
2. Then add the
product to the
numerator
3. The result is the
new numerator
(over the same
denominator)
2 x 4 = 8 8 + 3 = 11

17. 1. Properties of Numbers
Comparing Fractions
 Two methods
 Re-express them with a common
denominator.
 Convert them both to decimals.
 Use your calculator!!
13

18. #2
14
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

19. #2
14
Answer: D
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

20. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor Least Common Multiple
15

21. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
 Largest factor common to two or
more numbers
 Also called greatest common
divisor (gcd)
Least Common Multiple
15

22. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
 Largest factor common to two or
more numbers
 Also called greatest common
divisor (gcd)
 How to find it:
 Write out a list of factors for
each number – find the largest
in common
 Use your calculator:
Least Common Multiple
15

23. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
 Largest factor common to two or
more numbers
 Also called greatest common
divisor (gcd)
 How to find it:
 Write out a list of factors for
each number – find the largest
in common
 Use your calculator:
Least Common Multiple
15
MATH  NUM  GCD(

24. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
 Largest factor common to two or
more numbers
 Also called greatest common
divisor (gcd)
 How to find it:
 Write out a list of factors for
each number – find the largest
in common
 Use your calculator:
Least Common Multiple
 Smallest number that is a multiple
of two or more numbers
15
MATH  NUM  GCD(

25. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
 Largest factor common to two or
more numbers
 Also called greatest common
divisor (gcd)
 How to find it:
 Write out a list of factors for
each number – find the largest
in common
 Use your calculator:
Least Common Multiple
 Smallest number that is a multiple
of two or more numbers
 How to find it:
 Write out a list of multiples for
each number – find the smallest
number that appears on both
(all) lists
 Use your calculator:
15
MATH  NUM  GCD(

26. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
 Largest factor common to two or
more numbers
 Also called greatest common
divisor (gcd)
 How to find it:
 Write out a list of factors for
each number – find the largest
in common
 Use your calculator:
Least Common Multiple
 Smallest number that is a multiple
of two or more numbers
 How to find it:
 Write out a list of multiples for
each number – find the smallest
number that appears on both
(all) lists
 Use your calculator:
15
MATH  NUM  LCM(MATH  NUM  GCD(

27. #3
16
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

28. #3
16
Answer: C
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

29. 2. Counting Consecutive Integers
To count consecutive integers, simplify the problem to
identify a pattern (or equation!), and then apply it to
the larger context.
17

30. 2. Counting Consecutive Integers
To count consecutive integers, simplify the problem to
identify a pattern (or equation!), and then apply it to
the larger context.
17
Example 1:
How many even integers are there between 3 and 525?

31. #4
18
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

32. #4
18
Answer: E
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

33. 3. Number Line
 Can be asked to locate rational numbers on the number line or find the
distance between points on the number line
19

34. 3. Number Line
 Can be asked to locate rational numbers on the number line or find the
distance between points on the number line
 Draw/label a number line as you work through these questions!
19

35. 3. Number Line
 Can be asked to locate rational numbers on the number line or find the
distance between points on the number line
 Draw/label a number line as you work through these questions!
19
Example:
What is a irrational number between points A and B on the
number line?

36. 3. Number Line
 Can be asked to locate rational numbers on the number line or find the
distance between points on the number line
 Draw/label a number line as you work through these questions!
19
Example:
What is a irrational number between points A and B on the
number line?
0 1 2-1-2
A B C
𝐴. − 3 𝐵. −
1
2
2 C. −2 D. 3 E. 2

37. #5
20
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

38. #5
20
Answer: C
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

39. 4. Radicals
Simplifying Square Roots
21

40. 4. Radicals
Simplifying Square Roots
1. Write the number under the square root sign as a product of a perfect
square and another number
• Will be beneficial to find the largest perfect square factor
2. Take the square root of the perfect square and take it outside the
radical sign
21

41. 4. Radicals
Simplifying Square Roots
1. Write the number under the square root sign as a product of a perfect
square and another number
• Will be beneficial to find the largest perfect square factor
2. Take the square root of the perfect square and take it outside the
radical sign
21

42. 4. Radicals
Simplifying Square Roots
1. Write the number under the square root sign as a product of a perfect
square and another number
• Will be beneficial to find the largest perfect square factor
2. Take the square root of the perfect square and take it outside the
radical sign
21
If you don’t find the largest
perfect square factor, you can
still get the right answer. It will
just take longer!

43. 4. Radicals
Multiplying and Dividing Roots
 Recall that fractional exponents indicate a root! The same
rules apply with fractional exponents.
22

44. #6
23
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

45. #6
23
Answer: D
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

46. I. Number & Quantity
24
A. Real Number System
B. Complex Number System
C. Vectors
D. Matrices

47. The Complex Number System
 Imaginary number: any number that involves the square root of a
negative number
 Ex: ,
 Basic definition to remember:
 Complex Plane: a 2D space that represents both real and imaginary
numbers
25

48. The Complex Number System
 Imaginary number: any number that involves the square root of a
negative number
 Ex: ,
 Basic definition to remember:
 Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Simplification

49. The Complex Number System
 Imaginary number: any number that involves the square root of a
negative number
 Ex: ,
 Basic definition to remember:
 Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Exponent Pattern
Simplification

50. The Complex Number System
 Imaginary number: any number that involves the square root of a
negative number
 Ex: ,
 Basic definition to remember:
 Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Exponent Pattern
Simplification
= i

51. The Complex Number System
 Imaginary number: any number that involves the square root of a
negative number
 Ex: ,
 Basic definition to remember:
 Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Exponent Pattern
Simplification
= i

52. The Complex Number System
 Imaginary number: any number that involves the square root of a
negative number
 Ex: ,
 Basic definition to remember:
 Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Exponent Pattern
Simplification
= –i
= –i
= i

53. The Complex Number System
 Imaginary number: any number that involves the square root of a
negative number
 Ex: ,
 Basic definition to remember:
 Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Exponent Pattern
Simplification
= –i
= –i
= i

54. #7
26
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

55. #7
26
Answer: C
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

56. I. Number & Quantity
27
A. Real Number System
B. Complex Number System
C. Vectors
D. Matrices

57. 1. What is a Vector?
 A quantity that has both a
magnitude and a direction
 Vectors have a component
form to indicate the magnitudes
in each direction
 Ex: <6, -5>
28

58. 1. What is a Vector?
 A quantity that has both a
magnitude and a direction
 Vectors have a component
form to indicate the magnitudes
in each direction
 Ex: <6, -5>
28
Vector’s X
Magnitude

59. 1. What is a Vector?
 A quantity that has both a
magnitude and a direction
 Vectors have a component
form to indicate the magnitudes
in each direction
 Ex: <6, -5>
28
Vector’s X
Magnitude
Vector’s Y
Magnitude

60. 2. Unit Vector Notation
29

61. #8
30
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

62. #8
30
Answer: D
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

63. I. Number & Quantity
31
A. Real Number System
B. Complex Number System
C. Vectors
D. Matrices

64. 1. What is a matrix?
 A matrix is a rectangular
organization of a set of
numbers into rows and
columns
 Nomenclature
 m x n matrix where m is
the number of rows and
n is the number of
columns
 Think RC Cola – rows
first, columns second
32
This is a 3 x 4 matrix
Column #1
Row #3

65. 1. What is a matrix?
 A matrix is a rectangular
organization of a set of
numbers into rows and
columns
 Nomenclature
 m x n matrix where m is
the number of rows and
n is the number of
columns
 Think RC Cola – rows
first, columns second
32
This is a 3 x 4 matrix
Column #1
Row #3
Where are matrices used?
Matrices are used in many mathematical
areas, including the following:
1. Solving complex systems of equations
2. In the Google (Search Engine) PageRank
System
3. In services like Netflix that need to
organize large quantities of data

66. 2. Matrix Addition and Subtraction
 Must have two (or more) matrices that are identical in size
 Add (or subtract) number in corresponding positions
33

67. 2. Matrix Addition and Subtraction
 Must have two (or more) matrices that are identical in size
 Add (or subtract) number in corresponding positions
33

68. 3. Scalar Multiplication
 Scalar Multiplication: multiplication of an entire matrix by
some factor
 Simply multiply each term by the given factor (aka
scalar)
34

69. 3. Scalar Multiplication
 Scalar Multiplication: multiplication of an entire matrix by
some factor
 Simply multiply each term by the given factor (aka
scalar)
34

70. 4. Matrix Multiplication
35

71. 4. Matrix Multiplication
 Must have two matrices (m x n) and (a x b) such that n
and a are the same
35

72. 4. Matrix Multiplication
 Must have two matrices (m x n) and (a x b) such that n
and a are the same
35
Must be the same

73. 4. Matrix Multiplication
 Must have two matrices (m x n) and (a x b) such that n
and a are the same
 Resulting matrix will be m x b
35
Must be the same

74. 4. Matrix Multiplication
 Must have two matrices (m x n) and (a x b) such that n
and a are the same
 Resulting matrix will be m x b
 Multiply each piece of the first row in Matrix #1 by the first
column in Matrix #2. Find the sum, and this goes in Row
1, Column 1 of the new matrix.
35
Must be the same

75. 4. Matrix Multiplication
 Must have two matrices (m x n) and (a x b) such that n
and a are the same
 Resulting matrix will be m x b
 Multiply each piece of the first row in Matrix #1 by the first
column in Matrix #2. Find the sum, and this goes in Row
1, Column 1 of the new matrix.
35
Must be the same

76. 5. Matrix Determinant
 Determinant is a quantity related to a square matrix
 Determinant of Matrix A is often called det(A)
 Sometimes (but not always!) the definition of determinant
is provided if it’s needed on the test
 Generally only used on ACT for 2 x 2 matrices
Example
Find det(A) where .
36

77. #9
37
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

78. #9
37
Answer: A
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

79. #10
38
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

80. #10
38
Answer: B
Calculator?
Answer Choice Approach?
Drawing?
Hypothetical Numbers?

81. List of Terms
 Real & Complex Number Systems
 Real numbers
 Complex numbers
 Integers
 Whole numbers
 Natural/Counting numbers
 Complex Plane
 Properties of Numbers
 Prime numbers
 Least Common Multiple/Least Common Denominator
 Greatest Common Factor
 Matrices
 Determinant
 Scalar
 Matrix product
39