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

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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

1.1 Numbers The real number line:
Real numbers: {xx is a rational or an irrational number} -3 -2 -1 1 2 3

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?

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

1.2 Fundamental Operations of Algebra
Adding numbers on the number line ( ): -4 -3 -2 -1 1 2 3 4 -2 -2

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

1.2 Fundamental Operations of Algebra
Subtraction: To subtract signed numbers: Change the subtraction to adding the number with the opposite sign

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,

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:

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,

1.2 Fundamental Operations of Algebra

1.2 Fundamental Operations of Algebra
Distributive property

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

1.3 Calculators and Approximate Numbers
Significant Digits – What’s the pattern? Number Significant Digits 4.537 4 2 70506 5 40.500

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

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

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.

1.4 Exponents Power Rule (a) for exponents:
Power Rule (b) for exponents: Power Rule (c) for exponents:

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

1.4 Exponents Changing from negative to positive exponents:
This formula is not specifically in the book but is used often:

1.4 Exponents Quotient rule for exponents:

1.4 Exponents A few tricky ones:

1.4 Exponents Formulas and non-formulas:     

1.4 Exponents Examples (true or false):

1.4 Exponents Examples (true or false):

1.4 Exponents Putting it all together (example):

1.4 Exponents Another example:

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

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

1.5 Scientific Notation Converting to scientific notation (examples):
Converting back – just undo the process:

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

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:

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:

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

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

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

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:

1.7 Adding and Subtracting Algebraic Expressions
Example:

1.8 Multiplication of Algebraic Expressions
Multiplying a monomial and a polynomial: use the distributive property to find each product. Example:

1.8 Multiplication of Algebraic Expressions
Multiplying two polynomials:

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

1.8 Multiplication of Algebraic Expressions
Squaring binomials: Examples:

1.8 Multiplication of Algebraic Expressions
Product of the sum and difference of 2 terms: Example:

1.9 Division of Algebraic Expressions
Dividing a polynomial by a monomial: divide each term by the monomial

1.9 Division of Algebraic Expressions
Dividing a polynomial by a polynomial:

1.9 Division of Algebraic Expressions
Synthetic division: answer is: remainder is: -1

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)

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:

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:

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

1.11 Formulas and Literal Equations
Example: d = rt; (d = 252, r = 45) then 252 = 45t divide both sides by 45:

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:

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.

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

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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