📌 Integers: Opposites

Q: What is an additive inverse?

An additive inverse of a number is another number that, when added to it, yields zero. For example, the additive inverse of 7 is -7 because 7 + (-7) = 0.

Q: Why does multiplying two negative numbers result in a positive number?

Multiplying two negative numbers yields a positive result because multiplying by a negative number flips the sign on the number line. Reversing a negative direction twice turns it back into a positive direction (e.g., $(-1) \times (-1) = 1$).

Q: How do you know if a rational number becomes a terminating or repeating decimal?

Perform long division ($p \div q$) and track the remainders: if long division reaches a remainder of $0$, it is a **terminating decimal** (e.g., $\frac{3}{8} = 0.375$). If a remainder repeats, the quotient digits repeat indefinitely (**repeating decimal**, e.g., $\frac{5}{11} = 0.\overline{45}$). *(Shortcut: A simplified fraction terminates if its denominator $q$ contains only $2$'s and/or $5$'s as prime factors.)*

Master opposite quantities, additive inverses (p + (-p) = 0), and subtraction on number lines.

Opposites

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Learning Topics

Learning Guide

1. The Number Line Axis

A number line is a single horizontal axis where every point corresponds to a real number.

The Origin ( $0$ ): Zero is the central reference point on the axis.
Positive Numbers ( $> 0$ ): Located to the right of zero ( $1, 2, 3, \dots$ ). As you move right, numbers get larger.
Negative Numbers ( $< 0$ ): Located to the left of zero ( $-1, -2, -3, \dots$ ). As you move left, numbers get smaller.

The Number Line: Positive & Negative Directions

2. Opposites

Two numbers that are the same distance from zero in opposite directions on the number line are called opposites or additive inverses.

Additive Inverse Rule: For any number $p$ , its opposite is $-p$ , and their sum is always zero:
$p + (-p) = 0$
Real-World Example: A gain of $50 combined with a loss of $50 equals a net change of $0.
Number Line Representation: $p + q$ represents starting at position $p$ and moving a distance $|q|$ units:
- To the right if $q > 0$ (positive direction).
- To the left if $q < 0$ (negative direction).

Number Line: Opposites (-3 and +3)

The distance of -3 from zero is 3 units (left), and +3 is 3 units (right). Sum: 3 + (-3) = 0

3. Absolute Value & Distance from Zero

The absolute value of a number is its distance from zero on the number line, regardless of direction. Because distance is never negative, the absolute value of any number is always non-negative (

|x| \ge 0

Notation: Absolute value is written with vertical bars around the number: $|x|$ .
Opposites Property: Opposite numbers have the exact same distance from zero, so their absolute values are equal:
$|-a| = |a|$

Examples: $|-5| = 5$ and $|+5| = 5$ .
Evaluating Expressions with Absolute Value:
Treat absolute value bars $| \dots |$ like grouping symbols (parentheses). First evaluate the expression inside the bars, then take the non-negative absolute value:
Example: Evaluate $|-18 + 7| - |-5|$ :
1. Evaluate inside first: $-18 + 7 = -11$ .
2. Take absolute value: $|-11| = 11$ and $|-5| = 5$ .
3. Subtract: $11 - 5 = 6$ .

Number Line: Absolute Value as Distance from Zero (|-5| = 5 and |+5| = 5)

The distance from -5 to 0 is 5 units, and the distance from +5 to 0 is 5 units. Both |-5| and |+5| equal 5.

4. Subtraction as Adding the Opposite

Subtracting a number is mathematically identical to adding its additive inverse (its opposite):

Subtraction Rule:
$p - q = p + (-q)$
Distance on a Number Line:
The distance between any two rational numbers $p$ and $q$ on a number line is the absolute value of their difference:
$\text{Distance} = |p - q|$

Example: The distance between $-5$ and $3$ is $|-5 - 3| = |-8| = 8$ units.

5. Properties of Operations

You can use mathematical properties to reorder and simplify expressions with positive and negative rational numbers:

Commutative Property of Addition: $a + b = b + a$ . Changing the order of addends does not change the sum.
Associative Property of Addition: $(a + b) + c = a + (b + c)$ . Changing the grouping of addends does not change the sum.

Strategy Example: To evaluate

(-14) + 25 + (-11)

, reorder and group terms to combine negatives first:

(-14) + 25 + (-11) = 25 + \big[(-14) + (-11)\big] = 25 + (-25) = 0

💡 Mastering Integers Pro-Tip

Think of subtraction as adding the opposite! For example,

a - b

is the same as

a + (-b)

. On a number line, positive numbers move right and negative numbers move left.

Learning Topics

Frequently Asked Questions

What is an additive inverse?

An additive inverse of a number is another number that, when added to it, yields zero. For example, the additive inverse of 7 is -7 because 7 + (-7) = 0.

Why does multiplying two negative numbers result in a positive number?

Multiplying two negative numbers yields a positive result because multiplying by a negative number flips the sign on the number line. Reversing a negative direction twice turns it back into a positive direction (e.g., $(-1) \times (-1) = 1$ ).

How do you know if a rational number becomes a terminating or repeating decimal?

Perform long division ( $p \div q$ ) and track the remainders: if long division reaches a remainder of $0$ , it is a terminating decimal (e.g., $\frac{3}{8} = 0.375$ ). If a remainder repeats, the quotient digits repeat indefinitely (repeating decimal, e.g., $\frac{5}{11} = 0.\overline{45}$ ). (Shortcut: A simplified fraction terminates if its denominator $q$ contains only $2$ 's and/or $5$ 's as prime factors.)