Learning Objectives
- Understand the concept of negative numbers and integers
- Represent integers on a number line
- Order and compare integers
- Perform addition and subtraction of integers
Key Concepts
What Are Integers?
Integers are the collection of whole numbers and negative numbers: ..., -3, -2, -1, 0, 1, 2, 3, ... Positive integers are 1, 2, 3, ... and negative integers are -1, -2, -3, ... Zero is neither positive nor negative. We need negative numbers to represent situations like temperatures below zero, depths below sea level, or losses in business.
Integers on a Number Line
On a number line, negative numbers are placed to the left of zero and positive numbers to the right. The number increases as we move to the right. So -3 is less than -1, and -1 is less than 2. Every positive integer is greater than every negative integer.
Addition of Integers
When adding two positive integers, the result is positive: 3 + 5 = 8. When adding two negative integers, add their absolute values and put a negative sign: (-3) + (-5) = -8. When adding a positive and a negative integer, find the difference of their absolute values and give the sign of the larger absolute value: (-7) + 4 = -3, because 7 - 4 = 3 and 7 is bigger, so the answer is negative.
Subtraction of Integers
Subtracting an integer is the same as adding its additive inverse. The additive inverse of a is -a. So a - b = a + (-b). For example: 5 - (-3) = 5 + 3 = 8. And (-4) - 6 = (-4) + (-6) = -10.
Real-life Use of Integers
Integers are used to show temperature (above and below zero), altitude (above and below sea level), profit and loss in business, and scoring in games where you can gain or lose points.
Summary
Integers extend the number system to include negative numbers. They can be represented on a number line where numbers increase from left to right. Addition and subtraction of integers follow specific rules based on the signs of the numbers. Understanding integers is essential for describing quantities that can be above or below a reference point.
Important Terms
- Integers: The collection of ..., -3, -2, -1, 0, 1, 2, 3, ...
- Positive Integers: Numbers greater than zero (1, 2, 3, ...)
- Negative Integers: Numbers less than zero (-1, -2, -3, ...)
- Absolute Value: The distance of a number from zero on the number line (always positive)
- Additive Inverse: The number that when added gives zero; the additive inverse of a is -a
Quick Revision
- Zero is an integer but it is neither positive nor negative
- On the number line, left means smaller and right means larger
- Every positive integer is greater than every negative integer
- Adding two negative integers gives a negative integer
- Subtracting is the same as adding the additive inverse
- The absolute value of -5 is 5, and the absolute value of 5 is also 5
- -1 is the largest negative integer; there is no smallest negative integer