Operations of real numbers - vocabulare, roots, order of operations, table

Operations of real numbers table

VOCABULARY of real numbers

•  Total or sum is the answer to an addition problem; the numbers added are addends

EX: In 5 + 9 = 14, 5 and 9 are addends and 14 is the total or sum

•  Difference is the answer to a subtraction problem; the number subtracted is the subtrahend; the number from which the subtrahend is subtracted is the minuend

EX: In 25 - 8 = 17, 25 is the minuend, 8 is the subtrahend, and 17 is the difference

•  Product is the answer to a multiplication problem; the numbers multiplied are factors

EX: In 15 x 6 = 90, 15 and 6 are factors and 90 is the product

•  Quotient is the answer to a division problem; the number being divided is the dividend; the number that you are dividing by is the divisor; if there is a number remaining after the division process has been completed, that number is the remainder

EX: In 45 ÷ 5 = 9, which may also be written as 5√45 or 45/5 is the dividend, 5 is the divisor, and 9 is the quotient

•  Prime numbers are natural numbers greater than 1 having exactly two factors, itself and 1

EX: 7 is prime because the only two natural real numbers that multiply to equal 7 are 7 and 1; 13 is prime because the only two natural numbers that multiply to equal 13 are 13 and 1

•  Composite numbers are natural numbers that have more than two factors

EX: 15 is a composite number because 1, 3, 5, and 15 all multiply in some combination to equal 15; 9 is composite because 1, 3, and 9 all multiply in some combination to equal 9

•  The greatest common factor (GCF) or greatest common divisor (GCD) of a set of numbers is the greatest natural number that is a factor of each of the numbers in the set— that is, the greatest natural number that will divide into all of the numbers in the set without leaving a remainder

EX: The GCF of 12, 30, and 42 is 6 because 6 divides evenly into 12, 30, and 42 without leaving remainders

•  The least common multiple (LCM) of a set of numbers is the least natural number that can be divided (without remainders) by each of the numbers in the set

EX: The LCM of 2, 3, and 4 is 12 because although 2, 3, and 4 divide evenly into many numbers, including 48, 36, 24, and 12, the least is 12

•  The denominator of a fraction is the number on the bottom; it is the divisor of the indicated division of the fraction

EX: In 5/8, 8 is the denominator and also the divisor

•  The numerator of a fraction is the number on the top; it is the dividend of the indicated division of the fraction

EX: In 3/4, 3 is the numerator and also the dividend

FUNDAMENTAL THEOREM OF ARITHMETIC

•  The fundamental theorem of arithmetic states that every composite number can be expressed as a unique product of prime numbers

EX: 15 = (3)(5), where 15 is composite and both 3 and 5 are prime; 72 = (2)(2)(2)(3)(3), where 72 is composite and both 2 and 3 are prime; note that 72 also equals (8)(9), but this does not demonstrate the theorem because neither 8 nor 9 is a prime number

EXPONENTS AND POWERS

•  An exponent indicates the number of times the base is multiplied by itself—that is, used as a factor

EX: In 53, 5 is the base and 3 is the exponent; to simplify 53, evaluate (5)(5)(5), which is 125; note that the base, 5, was multiplied by itself three times

•  Squaring a number means to multiply the number by itself twice; EX: 7 squared = 72 = (7)(7) = 49

•  Cubing a number means to multiply the number by itself three times; EX: 4 cubed = 43 = (4)(4)(4) = 64

•  Raising a number to a power means to multiply the number by itself as many times as the power indicates

EX: 6 to the 5 th power = 65 = (6)(6)(6)(6)(6) = 7,776

ROOTS

•  The square root of a number is the number that, when multiplied by itself, equals the given number
EX: Because (8)(8) = 64, the square root of 64 is 8; this is written as √64 = 8

•  The cube root of a number is the number that, when multiplied by itself three times, equals the given number; the index 3 indicates the operation is a cube root

EX: Because (2)(2)(2) = 8, the cube root of 8 is 2; this is written as 3√8 = 2

•  Finding the nth root of a number is the number that, when multiplied by itself n times, equals the given number

EX: Because (5)(5)(5)(5) = 625, the 4th root of 625 is 5; this is written as 4√625 = 5

ORDER OF OPERATIONS

•  The order in which addition, subtraction, multiplication, and division are performed determines the answer; when simplifying mathematical expressions, follow this order:

1. Parentheses: Any operations contained in parentheses are done first, if there are any; this also applies to these enclosure symbols: { } and [ ]

2. Exponents: Exponent expressions are simplified second, if there are any

3. Multiplication and division of real numbers: These operations are done next in the order in which they are in the expression, going left to right; that is, if division comes first, going left to right, then it is done first

4. Addition and subtraction: These operations are done next in the order in which they are in the expression, going left to right; that is, if subtraction comes first, going left to right, then it is done first

EX: To simplify 23 - 42 + 2(6 - 1), first simplify the parentheses so that the expression becomes 23 - 42 + 2(5); next simplify the exponent so that the expression becomes 23 - 16 + 2(5); next perform the multiplication so that the expression becomes 23 - 16 + 10; because the expression has both addition and subtraction, perform those operations in order from left to right; the expression becomes 7 + 10 and finally 17

table operations of real numbers


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