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1.1 Numbers

1.1 Numbers. Classifications of Numbers. The real number line: Real numbers: {x x is a rational or an irrational number}. 1.1 Numbers. -3. -2. -1. 0. 1. 2. 3. 1.1 Numbers. Double negative rule : -(-x) = x

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1.1 Numbers

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  1. 1.1 Numbers • Classifications of Numbers

  2. The real number line: Real numbers:{xx is a rational or an irrational number} 1.1 Numbers -3 -2 -1 0 1 2 3

  3. 1.1 Numbers • Double negative rule:-(-x) = x • Absolute Value of a number x: the distance from 0 on the number line or alternativelyHow is this possible if the absolute value of a number is never negative?

  4. 3 > -3 means 3 is to the right on the number line 1 < 4 means 1 is to the left on the number line 1.1 Numbers -4 -3 -2 -1 0 1 2 3 4

  5. Adding numbers on the number line (-2 + -2): 1.2 Fundamental Operations of Algebra -4 -3 -2 -1 0 1 2 3 4 -2 -2

  6. 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

  7. 1.2 Fundamental Operations of Algebra • Subtraction: • To subtract signed numbers:Change the subtraction to adding the number with the opposite sign

  8. 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,

  9. 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:

  10. 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,

  11. 1.2 Fundamental Operations of Algebra • Commutative property (addition/multiplication) • Associative property (addition/multiplication)

  12. 1.2 Fundamental Operations of Algebra • Distributive property

  13. 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

  14. 1.3 Calculators and Approximate Numbers • Significant Digits – What’s the pattern?

  15. 1.3 Calculators and Approximate Numbers • Precision: • Meaning of the Last Digit: 56.5 V means the number of volts is between 56.45 and 56.55

  16. 1.3 Calculators and Approximate Numbers • Rounding to a number of significant digits

  17. 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.

  18. 1.4 Exponents • Power Rule (a) for exponents: • Power Rule (b) for exponents: • Power Rule (c) for exponents:

  19. 1.4 Exponents • Definition of a zero exponent: • Definition of a negative exponent:

  20. 1.4 Exponents • Changing from negative to positive exponents: • This formula is not specifically in the book but is used often:

  21. 1.4 Exponents • Quotient rule for exponents:

  22. 1.4 Exponents • A few tricky ones:

  23. 1.4 Exponents • Formulas and non-formulas:

  24. 1.4 Exponents • Examples (true or false):

  25. 1.4 Exponents • Examples (true or false):

  26. 1.4 Exponents • Putting it all together (example):

  27. 1.4 Exponents • Another example:

  28. 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

  29. 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

  30. 1.5 Scientific Notation • Converting to scientific notation (examples): • Converting back – just undo the process:

  31. 1.5 Scientific Notation • Multiplication with scientific notation (answers given without exponents): • Division with scientific notation:

  32. 1.6 Roots and Radicals • is the positive square root of a, andis 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:

  33. 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:

  34. 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

  35. 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

  36. 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

  37. 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:

  38. 1.7 Adding and Subtracting Algebraic Expressions • Example:

  39. 1.8 Multiplication of Algebraic Expressions • Multiplying a monomial and a polynomial: use the distributive property to find each product.Example:

  40. 1.8 Multiplication of Algebraic Expressions • Multiplying two polynomials:

  41. 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

  42. 1.8 Multiplication of Algebraic Expressions • Squaring binomials: • Examples:

  43. 1.8 Multiplication of Algebraic Expressions • Product of the sum and difference of 2 terms: • Example:

  44. 1.9 Division of Algebraic Expressions • Dividing a polynomial by a monomial:divide each term by the monomial

  45. 1.9 Division of Algebraic Expressions • Dividing a polynomial by a polynomial:

  46. 1.9 Division of Algebraic Expressions • Synthetic division:answer is:remainder is: -1

  47. 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)

  48. 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:

  49. 1.10 Solving Equations • Decimals - Multiply each term on both sides by the smallest power of 10 that gets rid of all the decimalsMultiply by 100:Cancel:Distribute:Subtract 5x:Subtract 50:Divide by 5:

  50. 1.10 Solving Equations • Example:Clear fractions:Combine like terms:Get variables on one side:Solve for x:

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