Integers are whole numbers that can be positive, negative, or zero, and they are used whenever quantities can go above or below a reference point. Temperatures, bank balances, elevations, and game scores all use integers to show direction as well as size. A number line makes integer operations easier to see because moving right means the value increases and moving left means the value decreases.
Zero is the neutral anchor that separates positive numbers from negative numbers.
Key Facts
- Adding a positive integer moves right on the number line: -3 + 5 = 2.
- Adding a negative integer moves left on the number line: 4 + (-6) = -2.
- Subtracting an integer means adding its opposite: a - b = a + (-b).
- Same signs add and keep the sign: -7 + (-4) = -11 and 6 + 3 = 9.
- Different signs subtract absolute values and keep the sign of the number with greater absolute value: -9 + 5 = -4.
- For multiplication and division, same signs give a positive result and different signs give a negative result: (-4)(-3) = 12 and 20 ÷ (-5) = -4.
Vocabulary
- Integer
- An integer is a whole number that is positive, negative, or zero.
- Number line
- A number line is a straight line that shows numbers in order from least to greatest.
- Opposite
- The opposite of a number is the number the same distance from zero on the other side of the number line.
- Absolute value
- Absolute value is a number's distance from zero, so it is never negative.
- Sign
- A sign tells whether a number is positive or negative.
Common Mistakes to Avoid
- Treating subtraction as always making a number smaller is wrong because subtracting a negative can increase the value, such as 3 - (-5) = 8.
- Adding signs before comparing absolute values is wrong for mixed signs because -8 + 3 requires subtracting 3 from 8 and keeping the negative sign.
- Forgetting that zero is neither positive nor negative is wrong because sign rules for positive and negative numbers do not make zero positive or negative.
- Using addition sign rules for multiplication is wrong because (-2) + (-3) = -5 but (-2)(-3) = 6.
Practice Questions
- 1 Use a number line or integer rules to calculate -6 + 9.
- 2 Evaluate 12 - (-5) + (-8).
- 3 A student says that -4 + 7 must be negative because one number is negative. Explain whether the student is correct and justify your reasoning.