ISEE Middle Level

## Integers and the number line The **integers** are the whole numbers and their opposites: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$. On a number line the numbers increase to the right, so any number is greater than every number to its left. That means $-2 > -7$, because $-2$ sits further right, even though 7 is the larger digit. The **absolute value** of a number is its distance from 0, written $\lvert n \rvert$, so it is never negative. For example $\lvert -9 \rvert = 9$ and $\lvert 4 \rvert = 4$. Two numbers with the same absolute value and different signs are **opposites**, and they add to 0. To order integers, put the negatives first and ordered backwards by size: $-8 < -5 < -1 < 0 < 3$. The negative number with the larger absolute value is the smaller number. ## Adding and subtracting integers When the signs match, add the absolute values and keep the sign: $-6 + (-4) = -10$. When the signs differ, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value: $-7 + 12 = 5$ and $3 + (-9) = -6$. Subtracting a number is the same as adding its opposite, which turns every subtraction into an addition: $$ \begin{aligned} -3 - (-8) &= -3 + 8 = 5 \\ 5 - 11 &= 5 + (-11) = -6 \end{aligned} $$ ## Multiplying and dividing integers Two numbers with the same sign give a positive result, and two with different signs give a negative result. This holds for both multiplication and division. $$ (-4)(-6) = 24 \qquad (-5)(3) = -15 \qquad…