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**1.1 Numbers Classifications of Numbers Natural numbers {1,2,3,…}**

Whole numbers {0,1,2,3,…} Integers {…-2,-1,0,1,2,…} Rational numbers – can be expressed as where p and q are integers -1.3, 2, , Irrational numbers – not rational

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**1.1 Numbers The real number line:**

Real numbers: {xx is a rational or an irrational number} -3 -2 -1 1 2 3

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**1.1 Numbers Double negative rule: -(-x) = x**

Absolute Value of a number x: the distance from 0 on the number line or alternatively How is this possible if the absolute value of a number is never negative?

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**1.1 Numbers 3 > -3 means 3 is to the right on the number line**

1 < 4 means 1 is to the left on the number line -4 -3 -2 -1 1 2 3 4

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**1.2 Fundamental Operations of Algebra**

Adding numbers on the number line ( ): -4 -3 -2 -1 1 2 3 4 -2 -2

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**1.2 Fundamental Operations of Algebra**

Adding numbers with the same sign: Add the absolute values and use the sign of both numbers Adding numbers with different signs: Subtract the absolute values and use the sign of the number with the larger absolute value

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**1.2 Fundamental Operations of Algebra**

Subtraction: To subtract signed numbers: Change the subtraction to adding the number with the opposite sign

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**1.2 Fundamental Operations of Algebra**

Multiplication by zero: For any number x, Multiplying numbers with different signs: For any positive numbers x and y, Multiplying two negative numbers: For any positive numbers x and y,

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**1.2 Fundamental Operations of Algebra**

Reciprocal or multiplicative inverse: If xy = 1, then x and y are reciprocals of each other. (example: 2 and ½ ) Division is the same as multiplying by the reciprocal:

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**1.2 Fundamental Operations of Algebra**

Division by zero: For any number x, Dividing numbers with different signs: For any positive numbers x and y, Dividing two negative numbers: For any positive numbers x and y,

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**1.2 Fundamental Operations of Algebra**

Commutative property (addition/multiplication) Associative property (addition/multiplication)

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**1.2 Fundamental Operations of Algebra**

Distributive property

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**1.2 Fundamental Operations of Algebra**

PEMDAS (Please Excuse My Dear Aunt Sally) Parenthesis Exponentiation Multiplication / Division (evaluate left to right) Addition / Subtraction (evaluate left to right) Note: the fraction bar implies parenthesis

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**1.3 Calculators and Approximate Numbers**

Significant Digits – What’s the pattern? Number Significant Digits 4.537 4 2 70506 5 40.500

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**1.3 Calculators and Approximate Numbers**

Precision: Meaning of the Last Digit: V means the number of volts is between and 56.55 Number Precision 4.537 thousandths 56 units 56.00 hundredths 40.500

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**1.3 Calculators and Approximate Numbers**

Rounding to a number of significant digits Original Number Significant Digits Rounded Number 4.5371 1 5 2 4.5 3 4.54 4 4.537

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**1.3 Calculators and Approximate Numbers**

Adding approximate numbers – only as accurate as the least precise. The following sum will be precise to the tenths position.

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**1.4 Exponents Power Rule (a) for exponents:**

Power Rule (b) for exponents: Power Rule (c) for exponents:

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**1.4 Exponents Definition of a zero exponent:**

Definition of a negative exponent:

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**1.4 Exponents Changing from negative to positive exponents:**

This formula is not specifically in the book but is used often:

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1.4 Exponents Quotient rule for exponents:

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1.4 Exponents A few tricky ones:

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1.4 Exponents Formulas and non-formulas:

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1.4 Exponents Examples (true or false):

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1.4 Exponents Examples (true or false):

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1.4 Exponents Putting it all together (example):

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1.4 Exponents Another example:

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**1.5 Scientific Notation A number is in scientific notation if :**

It is the product of a number and a 10 raised to a power. The absolute value of the first number is between 1 and 10 Which of the following are in scientific notation? 2.45 x 102 12,345 x 10-5 0.8 x 10-12 -5.2 x 1012

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**1.5 Scientific Notation Writing a number in scientific notation:**

Move the decimal point to the right of the first non-zero digit. Count the places you moved the decimal point. The number of places that you counted in step 2 is the exponent (without the sign) If your original number (without the sign) was smaller than 1, the exponent is negative. If it was bigger than 1, the exponent is positive

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**1.5 Scientific Notation Converting to scientific notation (examples):**

Converting back – just undo the process:

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1.5 Scientific Notation Multiplication with scientific notation (answers given without exponents): Division with scientific notation:

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1.6 Roots and Radicals is the positive square root of a, and is the negative square root of a because If a is a positive number that is not a perfect square then the square root of a is irrational. If a is a negative number then square root of a is not a real number. For any real number a:

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1.6 Roots and Radicals The nth root of a: is the nth root of a. It is a number whose nth power equals a, so: n is the index or order of the radical Example:

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**1.6 Roots and Radicals The nth root of nth powers:**

If n is even, then If n is odd, then The nth root of a negative number: If n is even, then the nth root is an imaginary number If n is odd, then the nth root is negative

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**1.7 Adding and Subtracting Algebraic Expressions**

Degree of a term – sum of the exponents on the variables Degree of a polynomial – highest degree of any non-zero term

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**1.7 Adding and Subtracting Algebraic Expressions**

Monomial – polynomial with one term Binomial - polynomial with two terms Trinomial – polynomial with three terms Polynomial in x – a term or sum of terms of the form

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**1.7 Adding and Subtracting Algebraic Expressions**

An expression is split up into terms by the +/- sign: Similar terms – terms with exactly the same variables with exactly the same exponents are like terms: When adding/subtracting polynomials we will need to combine similar terms:

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**1.7 Adding and Subtracting Algebraic Expressions**

Example:

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**1.8 Multiplication of Algebraic Expressions**

Multiplying a monomial and a polynomial: use the distributive property to find each product. Example:

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**1.8 Multiplication of Algebraic Expressions**

Multiplying two polynomials:

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**1.8 Multiplication of Algebraic Expressions**

Multiplying binomials using FOIL (First – Inner – Outer - Last): F – multiply the first 2 terms O – multiply the outer 2 terms I – multiply the inner 2 terms L – multiply the last 2 terms Combine like terms

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**1.8 Multiplication of Algebraic Expressions**

Squaring binomials: Examples:

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**1.8 Multiplication of Algebraic Expressions**

Product of the sum and difference of 2 terms: Example:

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**1.9 Division of Algebraic Expressions**

Dividing a polynomial by a monomial: divide each term by the monomial

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**1.9 Division of Algebraic Expressions**

Dividing a polynomial by a polynomial:

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**1.9 Division of Algebraic Expressions**

Synthetic division: answer is: remainder is: -1

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1.10 Solving Equations 1 – Multiply on both sides to get rid of fractions/decimals 2 – Use the distributive property 3 – Combine like terms 4 – Put variables on one side, numbers on the other by adding/subtracting on both sides 5 – Get “x” by itself on one side by multiplying or dividing on both sides 6 – Check your answers (if you have time)

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1.10 Solving Equations Fractions - Multiply each term on both sides by the Least Common Denominator (in this case the LCD = 4): Multiply by 4: Reduce Fractions: Subtract x: Subtract 5:

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1.10 Solving Equations Decimals - Multiply each term on both sides by the smallest power of 10 that gets rid of all the decimals Multiply by 100: Cancel: Distribute: Subtract 5x: Subtract 50: Divide by 5:

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1.10 Solving Equations Example: Clear fractions: Combine like terms: Get variables on one side: Solve for x:

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**1.11 Formulas and Literal Equations**

Example: d = rt; (d = 252, r = 45) then 252 = 45t divide both sides by 45:

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**1.11 Formulas and Literal Equations**

Example: Solve the formula for B multiply both sides by 2: divide both sides by h: subtract b from both sides:

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**1.12 Applied Word Problems 1 – Decide what you are asked to find**

2 – Write down any other pertinent information (use other variables, draw figures or diagrams ) 3 – Translate the problem into an equation. 4 – Solve the equation. 5 – Answer the question posed. 6 – Check the solution.

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1.12 Applied Word Problems Example: The sum of 3 consecutive integers is 126. What are the integers? x = first integer, x + 1 = second integer, x + 2 = third integer

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1.12 Applied Word Problems Example: Renting a car for one day costs $20 plus $.25 per mile. How much would it cost to rent the car for one day if 68 miles are driven? $20 = fixed cost, $.25 68 = variable cost

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