Groups of Numbers Natural or counting numbers: 1, 2, 3, 4,... Whole numbers: 0, 1, 2, 3,... Integers: ...-3, -2, -1, 0, 1, 2, 3, Rational numbers: Integers and fractions. Irrational numbers: Cannot be written as fractions: or π. Prime numbers: Divisible only by 1 and itself: 2, 3, 5, 7, 11, 13, . . . . (0 and 1 are not prime or composite.) Composite numbers: Divisible by more than just 1: 4, 6, 8, 9, 10, 12, . . . . Properties of Addition and Multiplication Closure: All answers fall into original set. Commutative: Order does not make any difference: a + b = b + a, ab = ba. Associative: Grouping does not make any difference: (a + b) + c = a + (b + c), (ab)c = a(bc). Identity: 0 for addition, 1 for multiplication. Inverse: Negative for addition, reciprocal for multiplication. Order of Operations Work within parentheses ( ), brackets [ ], and braces { } from innermost and work outward. Simplify exponents and roots working from left to right. Do multiplication and division, whichever comes first left to right. Do addition and subtraction, whichever comes first left to right. Rounding Off Underline the place value to which you're rounding off. Look to the immediate right (one place) of your underlined place value. Identify the number (the one to the right). If it is 5 or higher, round your underlined place value up 1 and change all the other numbers to its right to zeros. If less than 5, leave your underlined place value as it is and change all the other numbers to the right to zeros. Decimals To add or subtract decimals, simply line up the decimal points and then add or subtract as usual. To multiply decimals, just multiply as usual and then count the total number of digits above the line that are to the right of all decimal points. Place the decimal point in your answer so that there are the same number of digits to the right of the decimal point as there are above the line. To divide decimals, if the number you're dividing by has a decimal, move the decimal to the right as many places as possible and then move it under the division sign just as many places (add zeros if necessary). Move the decimal up to your answer. Fractions To add or subtract fractions, you must have a common denominator. If two fractions have a common denominator (like fractions), you simply add or subtract the numerator and keep the same denominator. (For example, 1/5 + 2/5 = 3/5.) If two fractions do not have a common denominator (unlike fractions), find a lowest common denominator (LCD), change each of the fractions to equivalent fractions with the new denominator, and then add or subtract the numerators and keep the same denominator. (For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6 When subtracting mixed numbers, you may have to "borrow" from the whole number. When you borrow 1 from the whole number, the 1 must be changed to a fraction. To multiply fractions, simply multiply the numerators and then multiply the denominators. (For example, 2/3 × 1/5 = 2/15.) Reduce to lowest terms if necessary. To divide fractions, invert the second fraction and then multiply. (For example, 1/5 ÷ 1/4 = 1/5 × 4/1 = 4/5.)
Pop Quiz! Which of the following expressions represents the statement “a number, x, decreased by 15 is less than or equal to 40”? 15 – x ≤ 40 x + 15 ≤ 40 x – 15 ≤ 40 Start Quiz