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P1 Real Numbers. Warm-up. How many people: -Are in the class? -Have a brother? -Have a sister? -Have no siblings? Lets make a Venn Diagram of that information. Complex Numbers.
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Warm-up How many people: -Are in the class? -Have a brother? -Have a sister? -Have no siblings? Lets make a Venn Diagram of that information.
Complex Numbers • We will worry about them down the road. Pretty much they help us define numbers when we have to take the square root of a negative.
Real Numbers • Numbers that describe quantities in everyday life. • They consist of rational numbers and irrational numbers. • Ex: 2, π, 2/5, .7, .1204123….,
Rational vs. Irrational Numbers Rational A real number that can be written as the ratio p/q of two integers where q≠0 Fractions/Decimals -Terminating: ½ , ¼… -Repeating: 1/3 = .33333… Irrational Numbers that cannot be written as a ratio of two integers Fractions/Decimals -Non-Terminating -Non-Repeating Ex) , etc.
Other Subsets… • Natural Numbers {1, 2, 3, 4, 5…} (Counting Numbers) • Whole Numbers {0, 1, 2, 3, 4, 5…} (Natural Numbers including Zero) • Integers {…-3, -2, -1, 0, 1, 2, 3…} (Whole Numbers and their Opposites)
Peardeck • Give the most specific set for each type of number.
Interval Notation [ ] – Including boundary ( ) – Excluding boundary Write each inequality in interval notation & draw on a number line Ex. 1) Ex. 2)
Writing Inequalities Example 1) x is at least -1 Example 2) y is at most 4 Example 3) z is at least -6 and at most 0
Given any two numbers… There are 3 possibilities… They are equal One is larger than the other Or the other is larger than the first
Absolute Value Measures the distance between the number and the origin (magnitude) If a is a real number, then
Examples 1. Evaluate when: a) x >0 b) x <0
Examples 2. Evaluate when: a) x > -5 b) x < -5
Properties Distance between Two Points
Set up the expression for… • The distance between x and 5 is 10. • b is at most 5 units from a • c is at least d units from 7
Algebraic Expressions Variables (letters) and constants (numbers) with +, -, x, ÷, and exponents Example: Evaluating: Substitute the given value for each of the variables in the expression ~ Evaluate the above expression for x = -5
Basic Rules of Algebra Let a, b, and c be real numbers, variables, or algebraic expressions Property
Properties of Negation Let a and b be real numbers, variables, or algebraic expressions Property
Properties of Equality Let a, b, and c be real numbers, variables, or algebraic expressions
Properties of Zero Let a and b be real numbers, variables, or algebraic expressions.
Properties and Operations of Fractions Let a, b, c, and d be real numbers, variables, or algebraic expressions such that and
Prime Numbers Integer with exactly 2 factors; one and itself Fundamental Theorem of Arithmetic Every positive integer greater than 1 can be written as the product of prime numbers Example) 36