Integers
Understand opposites, additive inverses, and operating with positive & negative rational numbers
Integers Study Guide
Explore the fundamental rules of positive and negative numbers, opposites, additive inverses, and subtraction on a number line.
Practice Topics
1. Opposites
Master opposite quantities, additive inverses (p + (-p) = 0), and subtraction on number lines.
Learning Guide
1. The Number Line Axis
A **number line** is a single horizontal axis where every point corresponds to a real number.
โข **The Origin ($0$):** Zero is the central reference point on the axis.
โข **Positive Numbers ($> 0$):** Located to the **right** of zero ($1, 2, 3, \dots$). As you move right, numbers get larger.
โข **Negative Numbers ($< 0$):** Located to the **left** of zero ($-1, -2, -3, \dots$). As you move left, numbers get smaller.
โข **The Origin ($0$):** Zero is the central reference point on the axis.
โข **Positive Numbers ($> 0$):** Located to the **right** of zero ($1, 2, 3, \dots$). As you move right, numbers get larger.
โข **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**.
โข **Additive Inverse Rule:** For any number $p$, its opposite is $-p$, and their sum is always zero:
$$p + (-p) = 0$$
โข **Real-World Example:** A gain of \$50 combined with a loss of \$50 equals a net change of \$0.
โข **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).
โข **Additive Inverse Rule:** For any number $p$, its opposite is $-p$, and their sum is always zero:
$$p + (-p) = 0$$
โข **Real-World Example:** A gain of \$50 combined with a loss of \$50 equals a net change of \$0.
โข **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. Subtraction as Adding the Opposite
Subtracting a number is mathematically identical to adding its additive inverse (its opposite):
โข **Subtraction Rule:**
$$p - q = p + (-q)$$
โข **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.
โข **Subtraction Rule:**
$$p - q = p + (-q)$$
โข **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.
๐ก 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.