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2-7: Square Roots and Real Numbers © William James Calhoun, 2001 OBJECTIVE: You must be able to find a square root, classify numbers, and graph solution of inequalities on number lines. square root - one of two equal factors of a number A number that will multiply by itself to get another given number. perfect square - a rational number whose square root is a rational number For an example of these two terms, 9 * 9 = 81. 9 is the square root of 81 since 9 times itself yields 81. 81 is a perfect square since it is a rational number and its square root, 9, is a rational number. Your calculators should have a square root key. It looks something like this:
2-8: Square Roots and Real Numbers © William James Calhoun, 2001 If x 2 = y, then x is a square root of y. DEFINITION OF SQUARE ROOT radical sign - the symbol for square root There are three modes of square roots: To find a square root without a calculator, you ask yourself, “What times itself will get me this number?” What times itself gives you 16? Answer: 4, so indicates the principal square root of 81. indicates the negative square root of 81. indicates both square roots of 81. is read “plus or minus the square root of 81.”
2-7: Square Roots and Real Numbers © William James Calhoun, 2001 EXAMPLE 1: Find each square root. A. B. C. This represents the principal square root of 25. Since 5 2 = 25, you know the answer is: 5 This represents the negative square root of 144. Since 12 2 = 144, you know the answer is: -12 This represents both the positive and negative square roots of 144. 0.4 2 = 0.16, so: -12 Remember: The easy way to answer these problems is use your calculator to get the principal square root of the number, the put the sign from the problem on your answer. In fact, unless it is an easily-remembered perfect square (like 4, 16, 25, 144, etc.) you will need to use a calculator.
2-7: Square Roots and Real Numbers © William James Calhoun, 2001 EXAMPLE 2: Use a calculator to evaluate each expression if x = 2401, a = 147, and b = 78. A.B. Replace x with 2401. Use calculator: = 49 Replace a with 147 and b with 78. Combine like terms. Use calculator: = 15 Now from square roots with nice rational answers to square roots and other numbers which can not be written as fractions.
2-7: Square Roots and Real Numbers © William James Calhoun, 2001 Question: What is the value of ? Your calculator should give you 1.412136… Notice this decimal does not appear to terminate or repeat. The decimal continues indefinitely without repeating. This brings up some new options for our Number Sets. Remember the chart and Venn diagrams from earlier: Natural Numbers Whole Numbers Integers Venn Diagram Well, there is more, as evidenced on the next slide.
2-7: Square Roots and Real Numbers © William James Calhoun, 2001 Irrationals Rationals Integers Whole Numbers Natural Numbers Real Numbers
2-7: Square Roots and Real Numbers © William James Calhoun, 2001 EXAMPLE 3: Name the set or sets of numbers to which each real number belongs. A. 0.833333333… B. C. D. Rational Rational Integer Rational Integer Whole Natural Irrational This is a repeating decimal, so it is rational. It is not an integer, whole number, or natural number. The only answer is then: This simplifies to: -4 which can be written as a fraction, so it is rational. Also, -4 is one of the integers. The answer is: This simplifies to: 7 which can be written as a fraction, so it is rational. Also, 7 is an integer, a whole number, and a natural number. The answer is: Plus this into a calculator. The result is: 10.95445115… which is non-repeating and non-terminating, so there can be only one answer:
2-7: Square Roots and Real Numbers © William James Calhoun, 2001 EXAMPLE 3: The area of a square is 235 square inches. Find its perimeter to the nearest hundredth. First find the length of each side. Area of a square = (side) 2. So, side = square root of Area. 235in 2 One side is found by plugging in for A. Remember the perimeter of a square has the formula: We will simplify this with the answer: The perimeter is about 72.11 inches. P = 4s Now we switch gears and do some graphing of inequalities. P = 4s P = 4(18.02775638) P = 72.111026551
2-7: Square Roots and Real Numbers © William James Calhoun, 2001 Rules for graphing inequalities on a number line: 1) Use the initial rules for graphing points on a number line from Section 2.1. 2) For , > and <, we use an open circle to signify the point is not included. 3) For =, > and <, we use a closed circle to signify the point is included. 4) Greater than has an arrow to the right. Less than has an arrow to the left. Not equal goes in both directions. > so full circlenot equal so open circle greater than so to rightnot equal so both directions EXAMPLE 5: Graph each solution set. A. y > -7B. p 3 / 4 -7 3/43/4
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