Chapter 1 Foundations for Algebra Variable – a letter or symbol used to represent a value that can change. Constant – a value that does not change. Numerical expression - contains only constants and operations. Algebraic expression – contains variables, constants, and operations. Coefficient – a number multiplied by a variable: 4x; 3y (4 and 3 are coefficients)

Absolute value – the distance from zero to a number on a number line. Opposites – two numbers with a sum of zero. Additive inverses – a number and its opposite. Reciprocals – two numbers whose product is 1. Multiplicative inverses – a number and its reciprocal.

Adding Real Numbers If two numbers have the same sign, add their absolute values and use the sign of the numbers. NUMBERS: 3 + 6 = 9; -2 + -5 = -7 If two numbers have different signs, find the difference of their absolute values and use the sign of the number with the greater absolute value. NUMBERS: -5 + 4 = -1; 8 + (-2) = 6

Subtracting Real Numbers To subtract a number, add its opposite. Then follow the rules for addition of real numbers. NUMBERS: 3 – 8 = 3 + (-8) ALGEBRA: a – b = a + (-b)

Multiplying and Dividing Real Numbers If two numbers have the same sign, their product or quotient is positive. NUMBERS: 4*5 = 20; -15/-5 = 3 If two numbers have different signs, their product or quotient is negative. NUMBERS: 8(-4) = 32; -24/6 = -4

Properties of Zero The product of any number and 0 is 0. The quotient of 0 and any number is 0. Division by 0 is undefined. NUMBERS: 0 * 8 = 0; 0/5 = 0; 12/0 undefined ALGEBRA: 0 * a = 0; 0/a = 0 a/0 undefined

.1 Square root – a number that is multiplied to itself to form a product is called a square root of that product. Perfect square – a number whose positive square root is a whole number. Natural numbers (N) – the counting numbers: 1, 2, 3, … Whole numbers (W) – the natural numbers and zero: 0, 1, 2, 3, …

Integers (Z) – whole numbers and their opposites: 0, 1, -1, 2, -2, 3, -3… Rational numbers – can be expressed in the form a/b, where a and b are both integers and b ‡ 0; ½, 4/1, 13/2 Terminating decimals – have a finite number of digits: 3.5, 2.318, 5.0 Repeating decimals – one or more digits behind the decimal point repeat: 1.3333…,6.27777…, 0.18181818….. Irrational numbers – cannot be expressed as a/b, they include square roots of non-perfect squares and non-terminating, nonrepeating decimals: 0.10110111…, , etc.

REAL NUMBERS Rational numbers (Q) Natural numbers Integers (Z) Whole numbers (W) Natural numbers (N)

Properties of Addition and Multiplication COMMUTATIVE PROPERTY: You can add and multiply numbers in any order. NUMBERS: 3 + 8 = 8 + 3; 6 * 7 = 7 * 6 ALGEBRA: a + b = b + a; ab = ba ASSOCIATIVE PROPERTY: When adding or multiplying, you can group any of the numbers together. NUMBERS: 6 + (8 + 3) = (6 + 8) + 3; 3*(4*5) = (3 * 4) * 5 ALGEBRA: a + (b + c) = (a + b) +c; a(bc) = (ab)c

Distributive Property You can multiply a number by a sum or difference; or multiply by each number in the sum or difference and then add. The result is the same. NUMBERS: 3(4 + 5) = 3(4) + 3(5); 8(5-3 ) = 8(5) – 8(3) ALGEBRA: a(b + c) = a(b) + a(c); a(b – c) = a(b) – a(c)

Coordinate plane – formed by the intersection of the x and the y axis. Origin – the point at which the x and the y axis intersect. Ordered pair – consists of an x-coordinate and a y-coordinate and is written (x,y). Quadrants – the four sections of the coordinate plane formed by the x and the y axes. Input – a value that is substituted for the independent variable in a relation or function. Output – the result of substituting a value for a variable in a function.