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Door 3 · Grade 8

🔢 Integers & Absolute Value

Integers include positive numbers, negative numbers, and zero (no fractions or decimals)

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Operations on Integers

What are Integers?

Integers are whole numbers that can be positive, negative, or zero. They include all positive numbers \(\1, 2, 3, ...\\), all negative numbers \(\-1, -2, -3, ...\\), and zero \(\0\\). Integers do NOT include fractions or decimals.

The Integer Family:

• Positive integers: \(1, 2, 3, 4, ...\) (numbers to the right of zero on number line)
• Negative integers: \(-1, -2, -3, -4, ...\) (numbers to the left of zero on number line)
• Zero: \(0\) (neither positive nor negative)

Operations on Integers

OperationRuleExampleAdditionSame signs: Add and keep the sign
Different signs: Subtract and use sign of larger number\((-5) + (-3) = -8\)
\((-7) + 4 = -3\)SubtractionAdd the opposite: \(a - b = a + (-b)\)\(5 - 8 = 5 + (-8) = -3\)MultiplicationSame signs = Positive product
Different signs = Negative product\((-4) × (-2) = 8\)
\((-3) × 5 = -15\)DivisionSame…

Worked example

Problem: Example 1: Add: \((-12) + 8 + (-5) + 3\)

  1. - Step 1: Group positive integers: \(8 + 3 = 11\)
  2. - Step 2: Group negative integers: \((-12) + (-5) = -17\)
  3. - Step 3: Add the groups: \(11 + (-17)\)
  4. - Step 4: Different signs, subtract: \(17 - 11 = 6\)
  5. - Step 5: Larger number is negative, so answer is negative: \(-6\)

Answer: - Step 5: Larger number is negative, so answer is negative: \(-6\)

Properties of Integers

What are Properties of Integers?

Properties are rules that always hold true when performing operations on integers. These properties help us simplify calculations and solve problems more easily.

The Main Properties of Integers under Addition and Multiplication:

PropertyMeaningAddition ExampleMultiplication ExampleClosure PropertySum/product of two integers is always an integer\((-7) + 4 = -3\) (integer)\(5 × (-3) = -15\) (integer)Commutative PropertyChanging order doesn't change the result\((-3) + 5 = 5 + (-3) = 2\)\((-4) × 6 = 6 × (-4) = -24\)Associative PropertyChanging grouping doesn't change the result\([2 + (-3)] + 5 = 2 + [(-3) + 5] = 4\)\([2 × (-3)] × 4 = 2 × [(-3) × 4] = -24\)Distributive PropertyMultiplication distributes over addition\(a × (b + c) = a × b + a × c\)Example: \(-2 × (3 + 4) = -2 × 7 = -14\) or \((-2 × 3) + (-2 × 4) = -6 + (-8) = -14\)…

Worked example

Problem: Example 1: Verify the commutative property of addition for integers \((-15)\) and \(8\).

  1. - Step 1: Left side: \((-15) + 8 = -7\)
  2. - Step 2: Right side: \(8 + (-15) = 8 - 15 = -7\)
  3. - Step 3: Compare: \((-15) + 8 = -7\) and \(8 + (-15) = -7\)
  4. - Step 4: Both sides are equal, so commutative property holds

Answer: - Step 4: Both sides are equal, so commutative property holds

Absolute Value & Applications

What is Absolute Value?

The absolute value of an integer is its distance from zero on the number line, regardless of direction. It is always non-negative (zero or positive).

Symbol: Absolute value is written as \[ |a|\) (vertical bars on both sides)

Definition: \[ |a| = cases a & if a ≥ 0 \ -a & if a

Examples:

• \(|5| = 5 \] (distance from 0 to 5 is 5 units)
• \(|-3| = 3\) (distance from 0 to -3 is 3 units)
• \(|0| = 0\)

Key Properties of Absolute Value:

PropertyRuleExampleNon-negativity\(\≥ 0\)\(\-7\= 7 ≥ 0\)Identity\(\= 0\) if and only if \(a = 0\)\(\0\= 0\)Multiplicative\( × b\= \× \\)\(\(-3) × 4\= \-12\= 12\) and \(\-3\× \4\= 3 × 4 = 12\)Triangle inequality\( + b\≤ \+ \\)\(\5 + (-3)\= \2\= 2 ≤ 5 + 3 = 8\)
Applications in Daily Life:

• Temperature Changes: The absolute change in temperature ignores whether it increased or decreased
• Bank Balances: The…

Worked example

Problem: Example 1: Find the value of: \(|-12| + |5| - |-3|\)

  1. - Step 1: Find absolute value of each term: \(|-12| = 12\)
  2. - Step 2: \(|5| = 5\)
  3. - Step 3: \(|-3| = 3\)
  4. - Step 4: Substitute: \(12 + 5 - 3\)
  5. - Step 5: \(12 + 5 = 17\), then \(17 - 3 = 14\)

Answer: - Step 5: \(12 + 5 = 17\), then \(17 - 3 = 14\)