Topic 1.2Foundation Topic

Integers

Quick Reference: Sign Rules

Multiplication & Division

+×+=+

5 × 3 = 15

−×−=+

(−5) × (−3) = 15

+×−=−

5 × (−3) = −15

−×+=−

(−5) × 3 = −15

Powers of Negatives

Even power → Positive

(-3)² = 9, (-2)⁴ = 16

Odd power → Negative

(-3)³ = -27, (-2)⁵ = -32

Basic operations

Addition and Subtraction

Master the rules for adding and subtracting positive and negative integers. Same signs: add and keep the sign. Different signs: subtract and take the sign of the larger number.

Key Concepts & Rules

•Adding positive: move right on the number line
•Adding negative: move left on the number line
•Subtracting negative = adding positive: a - (-b) = a + b
•Same signs: Add absolute values, keep the sign
•Different signs: Subtract absolute values, take sign of larger
•Zero is the identity: a + 0 = a and a - 0 = a

Worked Examples: Addition and Subtraction

Work out: -5 + 12 - 7

-5→+12→-7=0

Start at -5, move +12 to 7, then -7 to 0

Step 1:

(- 5 + 12) = 7

(different signs: subtract, take sign of larger)

Step 2:

7 - 7 = 0

Answer: 0

Rule: Same signs → add, keep sign. Different signs → subtract, keep larger sign.

Calculate: 15 - (3 - 7)

Brackets first: 3 - 7 = -4

15 - (-4) = 15 + 4 = 19

Step 1 (Brackets):

3 - 7 = - 4

Step 2:

15 - (- 4) = 15 + 4

(minus negative = plus)

Step 3:

15 + 4 = 19

Key rule: Subtracting a negative is the same as adding a positive!

The temperature at midnight was -5°C. By noon, it had risen by 12°C. What was the temperature at noon?

-5°C+12°C=7°C

Start:

- 5^{\circ} C

Change: Risen by

12^{\circ} C

means add 12

Calculation:

- 5 + 12 = 7

Answer:

7^{\circ} C

at noon

Real-world application: Temperature changes use integer addition/subtraction

A diver is at -25 m. She ascends 10 m, then descends 8 m. What is her new depth?

-25m+10m=-15m

-15m-8m=-23m

Start: -25 m (25m below sea level)

Ascend 10m:

- 25 + 10 = - 15

Descend 8m:

- 15 - 8 = - 23

Answer: New depth is -23 m (23m below sea level)

Ascending = positive change, Descending = negative change

Sign rules

Multiplication and Division

Apply the sign rules for multiplication and division: same signs give positive results, different signs give negative results. Remember BIDMAS for order of operations.

Key Concepts & Rules

•Positive × Positive = Positive (a × b = ab)
•Negative × Negative = Positive (-a × -b = ab)
•Positive × Negative = Negative (a × -b = -ab)
•Division follows same sign rules as multiplication
•Even power of negative = positive: (-a)²ⁿ = a²ⁿ
•Odd power of negative = negative: (-a)²ⁿ⁺¹ = -a²ⁿ⁺¹

Worked Examples: Multiplication and Division

Find the value of: 4 × (-6) ÷ (-3)

+×+=+

5 × 3 = 15

−×−=+

(−5) × (−3) = 15

+×−=−

5 × (−3) = −15

−×+=−

(−5) × 3 = −15

Step 1:

4 \times (- 6) = - 24

(positive × negative = negative)

Step 2:

(- 24) \div (- 3) = 8

(negative ÷ negative = positive)

Answer: 8

Sign rule: Same signs → positive, Different signs → negative

Simplify: (-3) × (10 - 4) / ((-2) + 5)

Brackets first:

(10 - 4) = 6

(-2 + 5) = 3

Step 1 (Brackets):

(10 - 4) = 6

(- 2 + 5) = 3

Step 2:

\frac{(- 3) \times 6}{3} = \frac{- 18}{3}

Step 3:

- 18 \div 3 = - 6

Answer: -6

Remember BIDMAS: Brackets → Indices → Division/Multiplication → Addition/Subtraction

Calculate: (-2)³ × (-3)²

(-2)³

= (-2) × (-2) × (-2)

= -8

Odd power → negative

(-3)²

= (-3) × (-3)

= 9

Even power → positive

(-2)³:

(- 2) \times (- 2) \times (- 2) = 4 \times (- 2) = - 8

(-3)²:

(- 3) \times (- 3) = 9

Multiply:

(- 8) \times 9 = - 72

Answer: -72

Rule: Even power of negative → positive. Odd power of negative → negative.

If a × b = -24 and a > 0, what are the possible values of a and b?

Factor pairs of 24:

1 × 24

2 × 12

3 × 8

4 × 6

Since a > 0 and product is negative, b must be negative

Condition:

a \times b = - 24

and

a > 0

Sign analysis: Positive × negative = negative, so

b < 0

Possible pairs (a, b):

(1, -24), (2, -12), (3, -8), (4, -6)

(6, -4), (8, -3), (12, -2), (24, -1)

8 possible pairs where a is positive and b is negative

Powers and roots

Squares, Cubes, and Roots

Calculate squares, cubes, square roots, and cube roots of integers. Understand that even powers of negatives are positive while odd powers preserve the sign. Cube roots of negatives are negative.

Key Concepts & Rules

•Square: n² = n × n. Always non-negative
•Square root √n: Principal (non-negative) root
•If x² = a, then x = ±√a (two solutions)
•Cube: n³ = n × n × n. Preserves sign
•Cube root ∛n: Preserves sign (∛-8 = -2)
•If x³ = a, then x = ∛a (one real solution)

Worked Examples: Squares, Cubes, and Roots

Calculate: (-2)⁴ + ∛(-8)

(-2)⁴

= 16

Even power → +

∛(-8)

= -2

Cube root of -ve is -ve

16 + (-2) = 14

(-2)⁴:

(- 2) \times (- 2) \times (- 2) \times (- 2) = 16

∛(-8):

\sqrt[3]{- 8} = - 2

(since (-2)³ = -8)

Add:

16 + (- 2) = 14

Answer: 14

Even powers of negatives are positive. Cube roots preserve the sign.

Evaluate: √81 - √25

√81

= 9

√25

= 5

√81:

\sqrt{81} = 9

(since 9² = 81)

√25:

\sqrt{25} = 5

(since 5² = 25)

Subtract:

9 - 5 = 4

Answer: 4

Note: √ always means the positive (principal) square root

If a² = 64, find all possible values of a.

Both 8 and -8 squared give 64:

8²

= 8 × 8

= 64 ✓

(-8)²

= (-8) × (-8)

= 64 ✓

Equation:

a^{2} = 64

Taking square root:

a = \pm \sqrt{64}

Remember: Both 8² = 64 AND (-8)² = 64

Answer:

a = 8

a = - 8

When solving x² = a, always give BOTH positive and negative solutions!

If b³ = -125, find the value of b.

Cube root preserves the sign:

∛(-125)

= -5

since (-5)³ = -125

Equation:

b^{3} = - 125

Taking cube root:

b = \sqrt[3]{- 125}

Unlike squares: Cube roots of negatives are negative

Answer:

b = - 5

Only ONE real solution for cube roots (sign is preserved)

Number line

Ordering and Comparing

Compare integers using the number line. Numbers to the right are always greater. Solve simple equations involving integers and find integers within specific ranges.

Key Concepts & Rules

•Number line: right is greater, left is smaller
•Negative numbers are less than positive numbers
•-a > -b when a < b (closer to zero is greater)
•When solving equations, perform same operation on both sides
•Integers are whole numbers: ..., -2, -1, 0, 1, 2, ...
•Greatest integer less than x = floor function ⌊x⌋

Worked Examples: Ordering and Comparing

Insert < or > to make true: -2 _ -8

Number line rule: Numbers to the RIGHT are greater

Visual: -2 is to the right of -8

Think: -2°C is warmer than -8°C

Answer:

- 2 > - 8

Closer to zero = greater (for negative numbers)

Arrange in ascending order: -3, 5, 0, -10, 2

-10<-3<0<2<5

Ascending order: Smallest to largest (left to right on number line)

Smallest (leftmost): -10

Next: -3, then 0, then 2

Largest (rightmost): 5

Answer:

- 10 < - 3 < 0 < 2 < 5

Negative numbers are always less than positive numbers

Solve for x: 2x - 5 = -11

2x - 5 = -11

+5 +5

2x = -6

÷2 ÷2

x = -3

Step 1: Add 5 to both sides:

2 x = - 11 + 5 = - 6

Step 2: Divide both sides by 2:

x = - 6 \div 2 = - 3

Check:

2 (- 3) - 5 = - 6 - 5 = - 11

✓

Answer:

x = - 3

Treat negative numbers like any other when solving equations

What is the smallest integer greater than -4.5?

-5<-4.5<-4 ✓

Question: Find smallest integer

> - 4.5

Integers near -4.5: -5, -4, -3, -2...

Which are greater than -4.5? -4, -3, -2...

Smallest of these: -4

Answer: -4

-4 > -4.5 on the number line (-4 is to the right)