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Real Numbers

Real Numbers. Rational and Irrational. Let’s look at the relationships between number sets. Notice rational and irrational numbers make up the larger number set known as Real Numbers.

nasim-caldwell
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Real Numbers

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  1. Real Numbers Rational and Irrational

  2. Let’s look at the relationships between number sets. Notice rational and irrational numbers make up the larger number set known as RealNumbers

  3. A number represents the value or quantity of something… Like how much money you have.. Or how many marbles you have… Or how tall you are.As you may remember from earlier grades there are different types of numbers.

  4. Rational Numbers Fractions are rational numbers in disguise

  5. Rational number – the set of all numbers that can be written in the form of a/b, where a and b are integers and b ≠ 0. Because a fraction can be written as a decimal, any rational number a/b can be written in decimal form. When you divide the denominator of a fraction into the numerator, the result is either a terminating or repeating decimal.

  6. A terminating decimalhas a finite number of nonzero digits to the right of the decimal point, such as 0.25 or 0.10. A repeating decimalhas a string of one or more digits that repeat infinitely, such as 1.555555…, or 0.345634563456… These repeating decimals can be indicated by a bar over the repeating digits, 1.5 or 0.3456

  7. Write 0.375 as a fraction in simplest form.0.375 375/1000 = 3/8, so 0.375 = 3/8Write 1/3 as a decimal.Divide 1 by 3 and you will see how the process will repeat infinitely.0.333333333333

  8. A number line - is an infinitely long line whose points match up with the real number system.

  9. Here are the rational numbers represented on a number line.

  10. Integers The coldest temperature on record in the U.S. is -80° F, recorded in 1971 in Alaska

  11. Integers are used to represent real-world quantities such as temperatures, miles per hour, making withdrawals from your bank account, and other quantities. When you know how to perform operations with integers, you can solve equations and problems involving integers.

  12. By using integers, you can express elevations above, below, and at sea level. Sea level has an elevation of 0 feet. Badwater Basin in Utah is -282 below sea level, and Clingman’s Dome in the Great Smokey Mountains is +6,643above sea level.

  13. If you remember, the whole numbers are the counting numbers and zero:0, 1, 2, 3,…Integers - the set of all whole numbers and their opposites. This means all the positive integers and all the negative integers together.

  14. Opposites – two numbers that are equal distance from zero on a number line; also called additive inverse.The additive inverse property states that if you add two opposites together their sum is 0-3 + 3 = 0

  15. Integers increase in value as you move to the right along a number line. They decrease in value as you move to the left. Remember to order numbers we use the symbol < means “less than,” and the symbol > means “is greater than.”

  16. A number’s absolute value - is it’s distance from 0 on a number line. Since distance can never be negative, absolute values are always positive. The symbol || represents the absolute value of a number. This symbol is read as “the absolutevalue of.” For example |-3| = 3.

  17. Finding absolute value using a number line is very simple. You just need to know the distance the number is from zero. |5| = 5, |-6| = 6

  18. Compare, Use <, >, or =. 1) -32 □ 32 2) 26 □ |-26| 3) -8 □ -12 4) Graph the numbers -2, 3, -4, 5. and -1 on a number line. Then list the numbers in order from least to greatest. 5) The coldest temperature ever recorded east of the Mississippi is fifty-four degrees below zero in Danbury, Wisconsin, on January 24, 1922. Write the temperature as an integer. Lesson Quiz

  19. Integer Operations Rules for Integer Operations

  20. Adding IntegersWhen we add numbers with the same signs,1) add the absolute values, and2) write the sum (the answer) with the sign of the numbers.When you add numbers with different signs,1) subtract the absolute values, and2) write the difference (the answer) with the sign of the number having the larger absolute value.

  21. Try the following problems 1) -9 + (-7) = -16 2) -20 + 15 = -5 3) (+3) + (+5) = +8 4) -9 + 6 = -3 5) (-21) + 21 = 0 6) (-23) + (-7) = -30

  22. Subtracting IntegersYou subtract integers by adding its opposite.9 – (-3)9 + (+3) = +12-7 – (-5)-7 + (+5) = -2

  23. Try the following problems 1) -5 – 4 = -5 + (-4) = -9 2) 3 – (+5) = 3 + (-5) = -2 3) -25 – (+25) = -25 + (-25) = -50 4) 9 – 3 = 9 + (-3) = +6 5) -10 – (-15) = -10 + (+15) = +5

  24. Adding and Subtracting Real NumbersWhen there is a negative sign in front of an expression in parentheses, such as –(3 + 4), there are two methods that you can use to simplify the expression. The first method is to simplify within the parentheses and then apply the rule for the addition of opposite signs as shown here. Simplify 10 –(-5 + 3) 10 –(-5 + 3) = 10 – (-2) Work within parentheses. Add -5 and 3 to get -2. = 10 + 2 Write subtraction as addition. = 12

  25. A second method that can be used to simplify –(3 + 4) is based on the (Multiplicative Property of -1)For all real numbers a, -1(a) = -a, or the opposite of a.This property can be extended to more than one term with the parentheses.The Opposite of a Sum -(a + b) = -a – bThe Opposite of a Difference -(a – b) = -a + b = b – aSimplify –(- 3 + 5)-1(-3 + 5) = 3 – 5 = -2

  26. Try the following problems 1) -12 – (4 + 9) 2) -(2x + 3) 3) -(3 + 4) – (5 + 6) 4) -(x + 7) 5) -(3 + 4) – (5 + 6) + (2 – 5) 6) -[(-2 – 8) – (-5 + 11)] – [(1 – 10) –(-3-3)]

  27. Multiplying and Dividing IntegersIf the signs arethe same, the answer is positive.If the signs are different,the answer is negative.

  28. Try the following problems Think of multiplication as repeated addition. 3 · 2 = 2 + 2 + 2 = 6 and 3 · (-2) = (-2) + (-2) + (-2) = -6 1) 3 · (-3) =Remember multiplication is fast adding = 3 · (-3) = (-3) + (-3) + (-3) = -9 2) -4 · 2 =Remember multiplication is fast adding = -4 · 2 = (-4) + (-4) = -8

  29. Multiply the following

  30. Dividing Integers Multiplication and division are inverse operations. They “undo” each other. You can use this fact to discover the rules for division of integers. 4· (-2) = -8-4· (-2) = 8 -8÷ (-2) = 48÷ (-2) = -4 same sign positive different signs negative The rule for division is like the rule for multiplication.

  31. Try the following problems 1) 72 ÷ (-9) 72 ÷ (-9) Think: 72 ÷ 9 = 8 -8 The signs are different, so the quotient is negative. 2) -144 ÷ 12 -144 ÷ 12 Think: 144 ∕12 = 12 -12 The signs are different, so the quotient is negative. 3) -100 ÷ (-5) Think: 100 ÷ 5 = 20 -100 ÷ (-5) The signs are the same, so the quotient is positive.

  32. Divide the following

  33. Find the sum or difference 1) -7 + (-6) = 2) -15 + 24 + (-9) = Evaluate x + y for x = -2 and y = -15 3) 3 – 9 = 4) -3 – (-5) = Evaluate x – y + z for x = -4, y = 5, and z = -10 Find the product or quotient 1) -8 · 12 = 2) -3 · 5 · (-2) = 3) -75 ÷ 5 = 4) -110 ÷ (-2) = 5) The temperature in Bar Harbor, Maine, was -3 F. During the night, it dropped to be four times as cold. What was the temperature then? Lesson Quiz

  34. Dividing Real NumbersBecause you can multiply any two real numbers, and every nonzero real number has a multiplicative inverse, you can define division using multiplication and multiplicative inverse. Example 1 1 2 Try This 5 1

  35. Using Multiplication and division with other operationsYou have learned the order of operations and the rules for adding subtracting, multiplying, and dividing real numbers. You can now simplify many expressions that involve positive and negative numbers. Simplify Another

  36. Simplify 1 3 -5 4 Try This

  37. The Distributive Property and Combining Like TermsA monomial - is a real number or the product of a real number and a variable raised to a whole-number power.For example, 6, -4c, and 3x2 are monomials.When there is both a number and a variable in the product, the number is called the coefficient. In the monomials -4c and 3x2, -4 and 3 are coefficients. • Recall that a monomial with no visible coefficient, such as x, actually has a coefficient of 1. Similarly, -x has a coefficient of -1. Using the Distributive Property

  38. Simplify Example 1 Example 2 Example 3 Try This

  39. Simplify More Distributive Property

  40. When you have unlike terms in a sum or difference within parentheses, you may have to use the opposite of a sum or a difference when simplifying. Simplify Example 1 Try This

  41. Using the Distributive Property to multiply and divideYou can use the Distributive Property to simplify a product. The Distributive Property applies to both positive and negative numbers. Example 1 Simplify Example 2 More

  42. Example 3 Next Example

  43. Example 4 Try This Try This

  44. Rational Numbers Fractions and Decimals

  45. Rational numbers, numbers that can be written in the form a/b (fractions), with integers for numerators and denominators.Integers and certain decimals are rational numbers because they can be written as fractions.

  46. Hint: When given a rational number in decimal form (such as 2.3456) and asked to write it as a fraction, it is often helpful to “say” the decimal out loud using the place values to help form the fraction.

  47. Write each rational number as a fraction:

  48. Hint: When checking to see which fraction is larger, change the fractions to decimals by dividing and comparing their decimal values.

  49. Examples of rational numbers are: 6 or 6/1 can also be written as 6.0 -2 or -2/1 can also be written as -2.0 ½ can also be written as 0.5 -5/4 can also be written as -1.25 2/3 can also be written as .66 2/3 can also be written as 0.666666… 21/55 can also be written as 0.38181818… 53/83 can also be written as 0.62855421687… the decimals will repeat after 41 digits

  50. Examples: Write each rational number as a fraction: • 0.3 • 0.007 • -5.9 • 0.45

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