ALEKS Pre-Algebra Study Notes
1. Numbers and Operations
a) Whole Numbers
Definition: Whole numbers are numbers without fractions or decimals. Examples: 0, 1, 2,
3, …
Operations:
o Addition: Combining numbers.
Example: 7 + 5 = 12
o Subtraction: Taking one number away from another.
Example: 12 − 7 = 5
o Multiplication: Repeated addition.
Example: 4 × 3 = 4 + 4 + 4 = 12
o Division: Splitting a number into equal parts.
Example: 12 ÷ 3 = 4
Tip: Addition and multiplication are commutative (order doesn’t matter), but subtraction and
division are not.
b) Integers
Definition: Integers include all whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2,
3…
Adding Integers:
o Same sign: add absolute values, keep the sign
Example: (−3) + (−5) = −8
o Different signs: subtract smaller from larger, keep the sign of the larger absolute
value
Example: 7 + (−5) = 2
Subtracting Integers: Change subtraction into adding the opposite
Example: 4 − (−3) = 4 + 3 = 7
Multiplying/Dividing Integers:
o Same signs → positive, different signs → negative
Example: (−3) × 4 = −12, (−12 ÷ −3) = 4
, c) Fractions
Definition: A fraction represents part of a whole: numerator/denominator (3/4)
Simplifying Fractions: Divide numerator and denominator by their GCF
Example: 6/8 → divide both by 2 → 3/4
Addition/Subtraction: Must have the same denominator
o Same denominator: 3/8 + 2/8 = 5/8
o Different denominators: find least common denominator (LCD)
Example: 1/4 + 2/3 → LCD = 12 → 3/12 + 8/12 = 11/12
Multiplication: Multiply numerators and denominators
Example: 2/3 × 4/5 = 8/15
Division: Multiply by the reciprocal of the second fraction
Example: (2/3) ÷ (4/5) = 2/3 × 5/4 = 10/12 = 5/6
d) Decimals
Definition: Fractions expressed with a decimal point
Operations:
o Addition/Subtraction: line up the decimal points
Example: 2.3 + 4.56 = 6.86
o Multiplication: multiply as whole numbers, then count decimal places
Example: 0.2 × 0.3 = 0.06
o Division: make the divisor a whole number, adjust the dividend accordingly
Example: 4.8 ÷ 0.6 → 48 ÷ 6 = 8
e) Percents
Definition: “Per 100”
o 25% = 25/100 = 0.25
Conversions:
o Fraction → Percent: (3/4) × 100 = 75%
1. Numbers and Operations
a) Whole Numbers
Definition: Whole numbers are numbers without fractions or decimals. Examples: 0, 1, 2,
3, …
Operations:
o Addition: Combining numbers.
Example: 7 + 5 = 12
o Subtraction: Taking one number away from another.
Example: 12 − 7 = 5
o Multiplication: Repeated addition.
Example: 4 × 3 = 4 + 4 + 4 = 12
o Division: Splitting a number into equal parts.
Example: 12 ÷ 3 = 4
Tip: Addition and multiplication are commutative (order doesn’t matter), but subtraction and
division are not.
b) Integers
Definition: Integers include all whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2,
3…
Adding Integers:
o Same sign: add absolute values, keep the sign
Example: (−3) + (−5) = −8
o Different signs: subtract smaller from larger, keep the sign of the larger absolute
value
Example: 7 + (−5) = 2
Subtracting Integers: Change subtraction into adding the opposite
Example: 4 − (−3) = 4 + 3 = 7
Multiplying/Dividing Integers:
o Same signs → positive, different signs → negative
Example: (−3) × 4 = −12, (−12 ÷ −3) = 4
, c) Fractions
Definition: A fraction represents part of a whole: numerator/denominator (3/4)
Simplifying Fractions: Divide numerator and denominator by their GCF
Example: 6/8 → divide both by 2 → 3/4
Addition/Subtraction: Must have the same denominator
o Same denominator: 3/8 + 2/8 = 5/8
o Different denominators: find least common denominator (LCD)
Example: 1/4 + 2/3 → LCD = 12 → 3/12 + 8/12 = 11/12
Multiplication: Multiply numerators and denominators
Example: 2/3 × 4/5 = 8/15
Division: Multiply by the reciprocal of the second fraction
Example: (2/3) ÷ (4/5) = 2/3 × 5/4 = 10/12 = 5/6
d) Decimals
Definition: Fractions expressed with a decimal point
Operations:
o Addition/Subtraction: line up the decimal points
Example: 2.3 + 4.56 = 6.86
o Multiplication: multiply as whole numbers, then count decimal places
Example: 0.2 × 0.3 = 0.06
o Division: make the divisor a whole number, adjust the dividend accordingly
Example: 4.8 ÷ 0.6 → 48 ÷ 6 = 8
e) Percents
Definition: “Per 100”
o 25% = 25/100 = 0.25
Conversions:
o Fraction → Percent: (3/4) × 100 = 75%
