Recap of integers

The integers are the set $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. They are the whole numbers extended to the negatives.

Idea

Integers stretch infinitely in both directions on the number line. Negatives sit to the left of $0$; positives to the right.

Comparing. On the number line, a number further to the right is greater. So $-2 > -5$, even though $5 > 2$. (Negative signs reverse the size order.)

Adding.

• Same sign: add absolute values, keep the sign. $(-3) + (-4) = -7$. $(+3) + (+4) = +7$.
• Different signs: subtract the smaller absolute value from the larger; take the sign of the larger. $(-7) + 3 = -4$ (because $7 > 3$ and $-7$ has larger magnitude).

Subtracting. Use the trick: $a - b = a + (-b)$. So $5 - (-3) = 5 + 3 = 8$. And $-4 - 6 = -4 + (-6) = -10$.

Absolute value. $|a|$ is the distance from $a$ to $0$, always non-negative. So $|-7| = 7$ and $|5| = 5$.

Real-life integers. Temperatures ($-10\,^\circ$C in Shimla in January), bank balances ($\rupee -500$ means owed), elevations ($-50$ m below sea level), sports scores (a "$-3$" handicap), etc.

Worked examples

Example 1. Compute $(-5) + 8$.

Different signs; $|8| > |-5|$; result has $+$ sign and magnitude $8 - 5 = 3$. So $+3$.

Example 2. Compute $-6 - (-4)$.

$-6 + 4 = -2$.

Example 3. Compute $-7 + (-3)$.

Same signs (both negative); add magnitudes: $7 + 3 = 10$. Keep sign: $-10$.

Example 4. What is $|-12| - |5|$?

$|-12| = 12$ and $|5| = 5$. Difference: $12 - 5 = 7$.

Try it yourself

1. Compute $(-9) + 4$.
2. Compute $(-6) + (-7)$.
3. Compute $10 - (-5)$.
4. Compute $-8 - 12$.
5. Which is larger, $-7$ or $-3$?
6. Find $|-15|$.
7. Arrange in increasing order: $-2, -5, 0, 3, -10$.
8. Temperature was $-3\,^\circ$C and rose by $7\,^\circ$C. New temperature?

Activity

Draw a number line from $-10$ to $10$ on graph paper. Mark $5$ integer pairs and compute their sums and differences by jumping along the line.