PROPERTIES OF REAL NUMBERS 1 ¾ .215 -7 PI

Subsets of real numbers – REVIEW Natural numbers numbers used for counting 1, 2, 3, 4, 5, …. Whole numbers the natural numbers plus zero 0, 1, 2, 3, 4, 5, … Integers the natural numbers ( positive integers ), zero, plus the negative integers

…,-4, -3, -2, -1, 0, 1, 2, 3, 4, … Rational numbers numbers that can be written as fractions decimal representations can either terminate or repeat Examples: fractions: 7/5 -3/2 -4/5 Any whole number can be written as a fraction by placing it over the number 1 8 = 8/1 100 = 100/1

terminating decimals ¼ = .25 2/5 = .4 Repeating decimals 1/3 = .3 2/3 = .6 These will always have a bar over the repeating section. Irrational numbers Cannot be written as fractions Decimal representations do not terminate or repeat

if the positive rational number is not a perfect square, then its square root is irrational Examples: Pi - non-repeating decimal 2 - not a perfect square

THE REAL NUMBERS Rational numbers Irrational numbers Integers Whole numbers Natural numbers

Graphing on a number line - 2 .3 -2 ¼ Tip: Best to put them as all decimals Put the square root in the calculator and find its equivalent -1.414… .333……… -2.25 -3 -2 -1 0 1 2 3

Ordering numbers Use the < , >, and = symbols Compare - .08 and - .1 Here again for square roots put them in the calculator and get their equivalents .08 = -.282842712475 - .1 = -.316227766017 So: - .1 < - .08 or - .08 > - .1

Properties of Real Numbers Opposite or additive inverse sum of opposites or additive inverses is 0 Examples: 400 4 1/5 - .002 - 4/9 -400 Additive inverse of any number a is -a 4/9 - 4 1/5 . 002

Reciprocal or multiplicative inverse product of reciprocals equal 1 Examples: 4 1/5 - .002 - 4/9 1/400 Multiplicative inverse of any number a is 1/a - 9/4 5/21 - 500

Other Properties: Addition: Closure a + b is a real number Commutative a + b = b + a 4 + 3 = 7` 3 + 4 = 7 numbers can be moved in addition Associative (a + b) + c = a + (b + c) (1 + 2) + 3 = 6 1+ (2 + 3) = 6 3 + 3 = 6 1 + 5 = 6 the order in which we add the numbers does not matter in addition

Identity a + 0 = a 7 + 0 = 7 when you add nothing to a number you still only have that number Inverse a + -a = 0 7 + -7 = 0

Multiplication Closure ab is a real number Commutative ab = ba 6(4) = 24 4 (6) = 24 When multiplying the numbers may be switched around, will not affect product Associative (ab)c = a(bc) The order in which they are multiplied does not affect the outcome of the product

(3*4)5 = 60 3(4*5) = 60 12(5) = 60 3(20) = 60 Identity a * 1 = a One times any number is the number itself 7 * 1 = 7 Inverse a * 1/a = 1 Product of reciprocals is one 7 * 1/7 = 7/7 = 1

DISTRIBUTIVE Property Combines addition and multiplication a(b + c) = ab + ac 2(3 + 4) = 2(3) + 2(4) 6 + 8 14

ABSOLUTE VALUE Absolute value is its distance from zero on the number line. Absolute value is always positive because distance is always positive Examples: -4 = 0 = -1 * -2 = 4 2

Assignment Page 8 – 9 Problems 34 – 60 even