Prerequisite Chapter Section 1 Real Numbers

Classifications of Numbers Imaginary Numbers will be introduced later.

Real Numbers Include any number that can be written as a decimal  Ex: Split into Rational and Irrational Numbers

Real Numbers Rational Numbers All the numbers that can be written as a ratio (fraction) This includes terminating and repeating decimals. Ex: 8, 10013, -54, 7/5, -3/25, 0, 0/6, -1.2,.09,.3333….

Real Numbers Rational Numbers Integers Negative and Positive Whole Numbers Zero Ex: -543, 76, 9, 0, -34

Real Numbers Rational Numbers Integers Whole Numbers Zero and positive integers Ex: 0, 1, 2, 3, 4, …

Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Also known as Counting Numbers Think of young children Ex: 1, 2, 3, 4, 5, 6, …

Real Numbers Irrational Numbers Numbers that cannot be written as ratios Decimals that never terminate and never repeat Square roots of positive non-perfect squares Ex: √2, -√7, √(8/11), …

Graphing Numbers on a Number Line Make sure your number lines have zero Make them fairly accurate Label Important points

Ordering Real Numbers Less Than<  Or equal to≤ Greater Than >  Or equal to ≥

Bounded Intervals * another way of writing inequalities [a, b] a < x < b closed (a, b) a < x < b open [a, b) a < x < b half-open (a, b] a < x < b half-open Use [ when is less/greater than or equal to Use ( when it is less/greater than

Unbounded Intervals [a, ∞) x > a closed (a, ∞) x > a open (-∞, b] x < b closed (-∞, b) x < b open

Evaluating Algebraic Expressions Variable – a symbol, usually a lowercase letter, that represents one or more numbers Algebraic expression or variable expression – a mathematical phrase that can include numbers, variables, and operation symbols Constant – letter or symbol used to represent a specific real number

Properties of Real Numbers Opposite or Additive Inverse – of any number a is –a  The sum of opposites is 0. Reciprocal or Multiplicative Inverse – of any number a is  Think of it as a flip Remember you may have to make a decimal into a fraction before flipping it. Use the place (hundredths) to write a fraction.  The product of reciprocals is 1.

Properties of Real Numbers PropertyAdditionMultiplication Closurea+b is a real number ab is a real number Commutative commute = to move a + b = b + aab = ba Associative associate = regroup (a+b)+c = a+(b+c)(ab)c = a(bc) Identitya+0=a,0+a=aa*1=a, 1*a=a Inversea+(-a)=0a*(1/a)=1,a≠0 Distributivea(b+c) = ab + ac

Properties of additive inverses 1. -(-u) = u 2. (-u)v = -uv 3. (-u)(-v) = uv 4. (-1)(u) = -u 5. -(u + v) = -u + -v Examples 1. -(-3) = 3 2. (-4)3 = (-6)(-7) = (-1)(5) = (7 + 9) = = -16

Exponents Properties of exponents 1. u m u n = u m + n 2. u m /u n = u m – n 3. u 0 = 1 4. u -n = 1/u n 5. (uv) m = u m v m 6. (u m ) n = u mn 7. (u/v) m = u m /v m

Scientific Notation c x 10 m where 1 < c < 10 and m is an integer Convert to scientific notation or standard notation X 10 8 = =

Examples Describe and graph the inequality 1. x < 3 all real numbers less than 3 2. Write an inequality and interval for the phrase: the real numbers between -4 and < x < -0.5 (-4, -0.5)

Examples 3. Convert to inequality notation: [-6, 3) -6 < x < 3 4. Convert to interval notation: x < 7 (-∞, 7] 5. Expand: (a + 2)x ax + 2x 6. Factor: 3y – by y(3 – b)

Example Simplify 1. (2ab 3 )(5a 2 b 5 ) = 2. = 3. =