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The Real Number System

The Real Number System. Rational Numbers. Irrational Numbers. Real Numbers (all numbers are real). …any number that is not rational. …-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5…. Natural Numbers. Example: = 3.14159…… e= 2.71828….. . Whole Numbers. Integers.

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The Real Number System

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  1. The Real Number System Rational Numbers Irrational Numbers Real Numbers (all numbers are real) …any number that is not rational …-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5… Natural Numbers • Example: • = 3.14159…… e= 2.71828….. Whole Numbers Integers rational number is a number that can be written as one integer over another: …-2/3, 5/1, 17/4….

  2. Inequality notation Read left to right … a<ba is lessthanba<b  a is less than or equal to ba >ba is greater thanba >b  a is greater than or equal to b a = b  a is not equal to b

  3. The Real Number Line Any real number corresponds to a point on the real number line. Order Property for Real Numbers Given any two real numbers a and b, - if a is to the left of b on the number line, then a < b. - if a is to the right of b on the number line, then a > b.

  4. Basic Properties of Real Numbers Did you ever notice when you perform SOME arithmetic operations the order which you perform them does not affect the answer? Example • Add: 4 + 3 = • Add: 3 + 4 = Commutative Property of Addition & Multiplication • Multiply: 4 x 3 = • Multiply: 3 x 4 = a + b = b + a a x b = b x a Think: Does this property apply to subtraction and division. Give an Example!

  5. Basic Properties of Real Numbers 2 x ( 4 + 3 ) = 2 x ( 7 ) = 14 Consider the following example solved in two ways: 2 x ( 4 + 3 ) = 2 x 4+2 x 3= 8 + 6 = 14 Same result Distributive Property of Real Numbers a (b + c) = ab+ac a (b – c ) = ab–ac

  6. Absolute Value of a Number • The absolute valueof x, notated | x |, measures theDISTANCEthat x is away from the origin(0) on the real number line.  Example: absolute value of 3

  7. is the reciprocal of 3 Clearly 3 is the reciprocal of The Reciprocal of a Number One number is the reciprocal of another if their product is 1. In general: The reciprocal of a fraction is obtained by interchanging the numerator and the denominator, i.e. by inverting the fraction. Example: The reciprocal of is

  8. Absolute Value of a Number Note:Distance is always going to be positive(unless it is0) whether the number you are taking the absolute value of is positive or negative. Example: absolute value of - 3

  9. Wrap up - Objectives • Identify what numbers belong to the set of • natural numbers, whole numbers, integers, rational numbers, • irrational numbers, and real numbers.  • Use the Order Property for Real Numbers. • Find the absolute value of a number.  • Write a mathematical statement with an equal sign or an • inequality. • Know and understand scientific notation.

  10. Fractions • A numeric FRACTION is a quotient of two numbers.  Numerator , denominator = 0 Fraction = Denominator Fraction Improper Fraction Proper Fraction Numerator is smaller than denominator Numerator is larger than denominator

  11. Fractions • Mixed numbers expression consisting of a whole number and a proper fraction: Convert mixed number to improper fraction: • Equivalent fractions fractions that represent the same number are called equivalent fractions.

  12. Fractions Reducing fractions Steps: • List the prime factors of the numerator and denominator. • Find the factors common to both the numerator and denominator and divide the numerator and denominator by all common factors (called canceling). • Reduce to the lowest terms. Example: Reduce to the lowest terms. Step 1 Step 2 Step 3

  13. Fractions • LCD – Least common denominator of two fractions: is the smallest number that can be divided by both denominators. Note: we need to find the LCD before we add or subtract two fractions. LCD of 3 and 5 = 15 (smallest number that can be divided by both 3 and 5)

  14. the factors here are 2 and 3 LCD = Fractions Find LCD and add the fractions 1. We write each denominator as a product of its prime factors. 1. 2. LCD = product of all factors taken at the biggest power

  15. Fractions • Write a fraction as a decimal: use a calculator to divide numerator by the denominator. If the division comes to an end , the decimal is a terminating decimal, if the division never ends the decimal is a repeating decimal. 1.Terminating decimal : 2. Repeating decimal: • Write a fraction as a percent: • A percent is a ratio of a number to 100. Percent means "per hundred." Thus, 20 • percent, or 20%, means . Fraction Decimal Percent x 100 = =

  16. Wrap up - Objectives • Know what the numerator and denominator of a fraction are. • Simplify a fraction. • Find the least common denominator of given fractions. • Multiply, divide, add and subtract fractions. • Write a fraction in decimal form and as a percent. • Know how to do percent increase and percent decrease word problems

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