Properties of integers - absolute value, comparing, substraction, addition, double negative

The set of integers includes the whole numbers and their opposites; some common uses of integers include temperatures (24°F and -2°C) and finance (the Smiths' net worth is -\$83,500, and the Johnsons' net worth is \$23,450)

The number line below illustrates the set of positive integers as well as the set of negative integers Properties of integers table

 ABSOLUTE VALUE of integers •  The absolute value of a number is the distance the number is away from 0 on a number line •  The absolute value of x is written in symbols as |x|; |x| = x if x > 0 or x = 0 and |x| = -x if x < 0; that is, the absolute value of a number is always the positive value of that number EX: |6| = 6 and |-6| = 6; the answer is positive 6 in both cases; this means the 6 and -6 are both 6 units away from 0 on a number line COMPARING and ORDERING INTEGERS •  A number line can be used to help compare and order integers; on a horizontal number line, integers to the right are greater than integers to the left; when comparing two integers, use the symbols < (less than) and > (greater than); the symbol points to the lesser number ADDITION of integers •  If the signs of the numbers are the same, add; the answer has the same sign as the numbersEX: (-4) + (-9) = -13 and 5 + 11 = 16 •  If the signs of the numbers are different, subtract; the answer has the sign of the number with the greater absolute value EX: (-4) + (9) = 5 and (4) + (-9) = -5 SUBTRACTION of integers •  Change subtraction to addition of the opposite number; a - b = a + (-b); that is, change the subtraction sign to addition and also change the sign of the number directly after the subtraction sign to the opposite; then follow the addition rules EX: (8) - (12) = (8) + (-12) = -4 and (-8) - (12) = (-8) + (-12) = -20 and (-8) - (-12) = (-8) + (12) = 4; note that the sign of the number in front of the subtraction sign never changes MULTIPLICATION and DIVISION •  Multiply or divide, then follow these rules to determine the sign of the answer: •  If the numbers have the same signs, the answer is positive •  If the numbers have different signs, the answer is negative •  It makes no difference which number is greater when you are trying to determine the sign of the answer EX: (-2)(-5) = 10 and (-7)(3) = -21 and (-18) + (6) = -3 DOUBLE NEGATIVE of integers •  -(-a) = a; that is, the sign in front of the parentheses changes the sign of the contents of the parentheses EX: -(-3) = 3 and -(3) = -3 •  -(a + b) = -a + (-b) and -(a - b) = -a - (-b) = -a + b; that is, when a negative sign is in front of a quantity, the negative sign is distributed to each term in the quantity EX: -5(a + 6) = -5a - 30 and -2(x - 3) = -2x + 6 RAISING A NEGATIVE TO A POWER •  When a negative number is raised to a power, the negative number must be in parentheses; the values -ax and (-a)x are not the same; in -ax, only a is raised to the power of x, and the final value is negative; in (-a)x, the final value is positive if x is even and negative if x is odd EX: -34 = -81 and (-3)4 = 81 