ASTB-E

Numbers form the foundation of all mathematical operations tested on the ASTB-E. Understanding number classifications allows you to recognize properties and apply appropriate rules during calculations.

Natural numbers are counting numbers beginning with 1 and continuing infinitely: 1, 2, 3, 4, 5, and so on. These numbers do not include zero, fractions, decimals, or negative values. Natural numbers are used when counting discrete objects.

Whole numbers include all natural numbers plus zero: 0, 1, 2, 3, 4, 5, and so on. The only difference between whole numbers and natural numbers is the inclusion of zero.

Integers expand the number system to include negative whole numbers. The set of integers includes: ...-3, -2, -1, 0, 1, 2, 3... Integers do not include fractions or decimals. Any whole number and its negative counterpart are integers.

Rational numbers are any numbers that can be expressed as a fraction p/q where p and q are integers and q is not zero. This category includes all integers (since any integer n can be written as n/1), all fractions, and all terminating or repeating decimals. Examples include 3/4, -2/5, 0.75, and 0.333... (repeating).

Irrational numbers cannot be expressed as a simple fraction. Their decimal representations neither terminate nor repeat. Common examples include π (pi, approximately 3.14159...), √2 (approximately 1.41421...), and e (approximately 2.71828...).

Real numbers encompass all rational and irrational numbers. Any number that can be placed on a number line is a real number. This includes all integers, fractions, decimals (terminating, repeating, and non-repeating), and irrational numbers.

A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. The first several prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Note that 2 is the only even prime number.

A composite number is a natural number greater than 1 that has more than two factors. Examples include 4, 6, 8, 9, 10, 12, and 15. The number 1 is neither prime nor composite.

Even numbers are integers divisible by 2 with no remainder. They end in 0, 2, 4, 6, or 8. Odd numbers are integers not divisible by 2. They end in 1, 3, 5, 7, or 9. Zero is considered even.

Properties to remember for calculations:

Even + Even = Even

Odd + Odd = Even

Even + Odd = Odd

Even × Even = Even

Odd × Odd = Odd

Even × Odd = Even

The order of operations determines the sequence for evaluating mathematical expressions. The standard order is represented by PEMDAS:

Parentheses (and other grouping symbols)

Exponents (and roots)

Multiplication and Division (left to right)

Addition and Subtraction (left to right)

Multiplication and division have equal priority and are performed left to right as they appear. The same applies to addition and subtraction.

First, evaluate the exponent: 4² = 16

Then multiply: 3 × 16 = 48 Then divide: 48 ÷ 2 = 24 Then add: 8 + 24 = 32

Finally subtract: 32: 5 = 27

The commutative property states that the order of numbers does not affect the result for addition and multiplication:

a + b = b + a

a × b = b × a

The associative property states that grouping does not affect the result for addition and multiplication:

(a + b) + c = a + (b + c)

(a × b) × c = a × (b × c)

The distributive property connects multiplication and addition:

a × (b + c) = (a × b) + (a × c)

The identity property establishes that adding 0 or multiplying by 1 leaves a number unchanged:

a + 0 = a

a × 1 = a

The inverse property establishes that adding a number's opposite or multiplying by its reciprocal yields the identity:

a + (-a) = 0 a × (1/a) = 1 (when a ≠ 0)

An exponent indicates how many times a base number is multiplied by itself. In the expression 5³, 5 is the base and 3 is the exponent, meaning 5 × 5 × 5 = 125.

Key exponent rules:

Negative exponent: a⁻ⁿ = 1/aⁿ

Product to power: (ab)ⁿ = aⁿbⁿ

Quotient to power: (a/b)ⁿ = aⁿ/bⁿ

A root is the inverse operation of an exponent. The square root of a number x is a value that, when multiplied by itself, equals x. The cube root is a value that, when multiplied by itself three times, equals x. √25 = 5 because 5 × 5 = 25 ³√27 = 3 because 3 × 3 × 3 = 27

Simplifying radicals involves factoring out perfect squares (or perfect cubes for cube roots): √72 = √(36 × 2) = √36 × √2 = 6√2 √50 = √(25 × 2) = 5√2

Radical rules:

√(ab) = √a × √b

√(a/b) = √a / √b

(√a)² = a

√a² = |a|

The absolute value of a number is its distance from zero on the number line, always expressed as a non-negative value. The absolute value of both 5 and -5 is 5, written as |5| = 5 and |-5| = 5.

When solving equations with absolute value, consider both positive and negative cases:

On a number line, positive numbers lie to the right of zero and negative numbers lie to the left. Numbers increase in value moving right and decrease moving left. The greater a negative number's absolute value, the smaller its actual value (for example, -10 is less than -3).

When adding numbers with the same sign, add their absolute values and keep the common sign: 5 + 8 = 13 (-5) + (-8) = -13

When adding numbers with different signs, subtract the smaller absolute value from the larger and use the sign of the number with the larger absolute value: 8 + (-5) = 3 (-8) + 5 = -3

To subtract a number, add its opposite: 7 - 10 = 7 + (-10) = -3 7 - (-10) = 7 + 10 = 17 (-7) - 10 = (-7) + (-10) = -17 (-7) - (-10) = (-7) + 10 = 3

The rules for multiplication and division are identical:

Positive × Positive = Positive

Negative × Negative = Positive

Positive × Negative = Negative

Negative × Positive = Negative

The same rules apply to division. When multiplying or dividing multiple numbers, count the negative signs. An even count of negatives yields a positive result; an odd count yields a negative result.