2.1 Real Numbers & Operations

The Real Number System

The real numbers include every number you can place on a number line: the integers (…, −3, −2, −1, 0, 1, 2, 3, …), the rational numbers (any fraction of two integers, such as 3/4 or −5), and the irrational numbers (non-repeating, non-terminating decimals such as π and √2). ALEKS PPL builds upward from this foundation, so fluency here protects every later score.

Key subsets to recognize: natural numbers (1, 2, 3, …), whole numbers (0, 1, 2, …), and integers (positive and negative whole numbers plus zero). On the test you will classify numbers and operate on them quickly, so memorize that every integer is rational, but not every rational number is an integer.

Order of Operations (PEMDAS)

The order of operations tells you which calculation to perform first so that everyone gets the same answer. Remember PEMDAS:

• P — Parentheses (and other grouping symbols)
• E — Exponents and roots
• MD — Multiplication and Division, left to right
• AS — Addition and Subtraction, left to right

The biggest trap: multiplication does not always come before division, and addition does not always come before subtraction. Within each pair you work strictly left to right.

Worked Example 1. Evaluate 6 + 2 × (3² − 1).

1. Parentheses first, but resolve the exponent inside: 3² = 9, so (9 − 1) = 8.
2. Multiplication: 2 × 8 = 16.
3. Addition: 6 + 16 = 22.

Worked Example 2. Evaluate 20 − 12 ÷ 4 + 3 × 2.

1. Division and multiplication left to right: 12 ÷ 4 = 3 and 3 × 2 = 6.
2. The expression becomes 20 − 3 + 6.
3. Add and subtract left to right: 20 − 3 = 17, then 17 + 6 = 23.

Signed Numbers and Absolute Value

Adding signed numbers: with the same sign, add the values and keep the sign (−5 + −3 = −8). With different signs, subtract the smaller value from the larger and keep the sign of the larger (−9 + 4 = −5). Subtracting means adding the opposite: 7 − (−2) = 7 + 2 = 9.

Multiplying and dividing signed numbers follows a simple sign rule:

So (−6)(−4) = 24, but (−6)(4) = −24, and −20 ÷ 5 = −4.

Absolute value |x| is the distance of x from zero, and distance is never negative: |−7| = 7 and |7| = 7. Watch the trap in −|−3|: the inner absolute value gives 3, and the outside negative makes it −3.

Worked Example 3. Evaluate −3 + |−8| × 2. The absolute value is 8, so 8 × 2 = 16, and −3 + 16 = 13.

Exponents, Roots, and Prime Factorization

An exponent is repeated multiplication: 2⁴ = 2 × 2 × 2 × 2 = 16. A square root undoes a square, so √49 = 7 because 7² = 49, and √64 = 8. Memorize the perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 for instant recognition.

Prime factorization rewrites a number as a product of primes. For 72: 72 = 8 × 9 = 2³ × 3². For 60: 60 = 2² × 3 × 5.

From prime factorizations you get two classic answers:

• GCF (Greatest Common Factor): multiply the lowest power of each shared prime.
• LCM (Least Common Multiple): multiply the highest power of every prime that appears.

Worked Example 4. Find the GCF and LCM of 60 and 72. Factor: 60 = 2² × 3 × 5 and 72 = 2³ × 3². Shared primes are 2 and 3. GCF = 2² × 3 = 12 (lowest powers). LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360 (highest powers). A quick check: GCF × LCM = 12 × 360 = 4320 = 60 × 72.

Common Mistakes

• Sign slips: forgetting that subtracting a negative adds, so 5 − (−4) = 9, not 1.
• Left-to-right errors: doing all multiplication before division instead of working left to right.
• GCF/LCM swap: using highest powers for the GCF; the GCF always uses the lowest shared powers.

Putting the Rules Together

Most ALEKS arithmetic items combine several of these ideas in a single expression, so a disciplined left-to-right pass through PEMDAS keeps you accurate under time pressure. The reliable routine is: first simplify everything inside grouping symbols, then evaluate exponents and roots, then handle multiplication and division as you read from left to right, and finally addition and subtraction the same way.

Worked Example 5. Evaluate (−2)³ + √36 − 4 × 2.

1. Exponent: (−2)³ = (−2)(−2)(−2) = −8. Note an odd power of a negative stays negative, while an even power such as (−2)² = 4 is positive.
2. Root: √36 = 6.
3. Multiplication: 4 × 2 = 8.
4. Combine left to right: −8 + 6 − 8 = −2 − 8 = −10.

Worked Example 6. Evaluate 18 ÷ (−3) + (−2)(−5) − |−4|.

1. Division of opposite signs: 18 ÷ (−3) = −6.
2. Product of two negatives: (−2)(−5) = 10.
3. Absolute value: |−4| = 4.
4. Combine: −6 + 10 − 4 = 4 − 4 = 0.

A final reminder on negatives and exponents: −3² and (−3)² are different. In −3² the exponent attaches only to the 3, so −3² = −(3 × 3) = −9, whereas (−3)² = 9 because the parentheses square the entire −3. ALEKS reliably tests this distinction, so read grouping symbols carefully before squaring.