Properties of real numbers, table

•  a + b is a real number; when you add two real numbers, the result is also a real number

EX: 3 and 5 are both real numbers; 3 + 5 = 8 and the sum, 8, is also a real number

•  a - b is a real number; when you subtract two real numbers, the result is also a real number

EX: 4 and 11 are both real numbers; 4 - 11 = -7 and the difference, -7, is also a real number

•  (a)·(b) is a real number; when you multiply two real numbers, the result is also a real number

EX: 10 and -3 are both real numbers; (10)(-3) = -30 and the product, -30, is also a real number

•  a/b is a real number when b ≠ 0; when you divide two real numbers, the result is also a real number unless the denominator (divisor) is 0

EX: -20 and 5 are both real numbers;  -20/5 = -4 and the quotient, -4, is also a real number

•  a + b = b + a; the order in which you add two numbers does not change the sum

EX: 9 + 15 = 24 and 15 + 9 = 24, so 9 + 15 = 15 + 9

•  (a)·(b) = (b)·(a); the order in which you multiply two numbers does not change the product

EX: (4)(26) = 104 and (26)(4) = 104, so (4)(26) = (26)(4)

•  a - b ≠ b - a; you cannot subtract in any order and get the same answer

EX: 8 - 2 = 6, but 2 - 8 = -6; there is no commutative property for subtraction

•  a/b ≠ b/a; you cannot divide in any order and get the same answer

EX: 8/2 = 4, but 2/8 = 0.25, so there is no commutative property for division

•  (a + b) + c = a + (b + c); the order in which you group more than two numbers when adding does not change the sum

EX: (2 + 5) + 9 = 7 + 9 = 16 and 2 + (5 + 9) = 2 + 14 = 16, so (2 + 5) + 9 = 2 + (5 + 9)

•  (a·b)·c = a·(b·c); the order in which you group more than two numbers when multiplying does not change the product

EX: (4 x 5) 8 = (20) 8 = 160 and 4 (5 x 8) = 4 (40) = 160, so (4 x 5) 8 = 4 (5 x 8)

•  The associative property does not work for subtraction or division

EX: For subtraction, (10 - 4) - 2 = 6 - 2 = 4, but 10 - (4 - 2) = 10 - 2 = 8; for division, (12 ÷ 6) ÷ 2 = (2) ÷ 2 = 1, but 12 ÷ (6 ÷ 2) = 12 ÷ 3 = 4; note that these answers are not the same

•  a + 0 = a; adding 0 to a number does not change its value

EX: 9 + 0 = 9 and 0 + 9 = 9

•  a·(1) = a; multiplying a number by 1 does not change its value

EX: 23· (1) = 23 and (1)· 23 = 23

•  There are no identities for subtraction or division

•  a + (-a) = 0; a number plus its additive inverse (the number with the opposite sign) is 0

EX: 5 + (-5) = 0 and (-5) + 5 = 0

•  a·(1/a) = 1; a number times its multiplicative inverse (the number written as a fraction and flipped) is 1

EX: 5·(1/5) = 1; the exception is 0 because 0 cannot be multiplied by any number and result in a product of 1

• a·(b + c) = a·b + a·c or a·(b - c) = a·b - a·c; each term in the parentheses must be multiplied by the term in front of the parentheses

EX: 4·(5 + 7) = 4·(5) + 4·(7) = 20 + 28 = 48

•  Sometimes variables are included within parentheses in mathematical expressions; using the distributive property allows you to remove the parentheses in such an expression

EX: 4·(5·a + 7) = 4·(5·a) + 4(7) = 20·a + 28

•  Reflexive: a = a; both sides of the equation are identical EX: 5 + k = 5 + k

•  Symmetric: If a = b, then b = a; this property allows you to exchange the two sides of an equation

EX: 4a - 7 = 9 - 7a + 15 becomes 9 - 7a + 15 = 4a - 7

•  Transitive: If a = b and b = c, then a = c; this property allows you to connect statements that are each equal to the same common statement

EX: 5a - 6 = 9k and 9k = a + 2; you can eliminate the common term 9k and write one equation: 5a - 6 = a + 2

•  Addition property of equality: If a = b, then a + c = b + c; this property allows you to add any number or algebraic term to any equation as long as you add it to both sides to keep the equation true

EX: 5 = 5; if you add 3 to one side and not the other, the equation becomes 8 = 5, which is false, but if you add 3 to both sides, you get a true equation (8 = 8); also, 5a + 4 = 14 becomes 5a + 4 + (-4) = 14 + (-4) if you add -4 to both sides; this results in the equation 5a = 10

•  Multiplication property of equality: If a = b, then ac = bc when c ≠ 0; this property allows you to multiply both sides of an equation by any nonzero value

EX: If 4a = -24, then (4a)(0.25) = (-24)(0.25) and a = -6; note that both sides of the equation were multiplied by 0.25