28: Square Roots and Real Numbers. 28: Square Roots and Real Numbers


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1 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 * is the square root of 81 since 9 times itself yields 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: x DEFINITION OF SQUARE ROOT If x 2 y, then x is a square root of y. radical sign  the symbol for square root There are three modes of square roots: 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. 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
2 EXAMPLE 1: Find each square root. A. 25 B. 144 C. ± This represents the principal square root of 25. Since , you know the answer is: 5 This represents the negative square root of 144. Since , you know the answer is: 12 This represents both the positive and negative square roots of , 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 easilyremembered perfect square (like 4, 16, 25, 144, etc.) you will need to use a calculator. EXAMPLE 2: Use a calculator to evaluate each expression if x 2401, a 147, and b 78. A. x B. ± a + b Replace x with Replace a with 147 ± and b with 78. Use calculator: 49 Combine like terms. ± 225 Use calculator: ± 15 Now from square roots with nice rational answers square roots and other numbers which can not be written as fractions. to 2
3 Question: What is the value of 2? Your calculator should give you 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: Sets Examples Natural numbers 1, 2, 3, 4, 5, Whole numbers 0, 1, 2, 3, 4, Integers, 2, 1, 0, 1, 2, Integers Well, there is more, as evidenced on the next slide. Whole Numbers Natural Numbers Venn Diagram Sets Examples Symbol Natural numbers 1, 2, 3, 4, 5, N Whole numbers 0, 1, 2, 3, 4, W Integers, 2, 1, 0, 1, 2, Z Numbers any number that can be written as a fraction includes repeating and terminating decimals Q Irrational Numbers numbers that cannot be written as fractions nonrepeating and nonterminating I Real Numbers the set of all and Irrational Number R s Real Numbers Integers Whole Numbers Natural Numbers Irrationals Real Numbers 3
4 EXAMPLE 3: Name the set or sets of numbers to which each real number belongs. A B. 16 C D. 120 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: Integer 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: Integer Whole Natural Plus this into a calculator. The result is: which is nonrepeating and nonterminating, so there can be only one answer: Irrational 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 s A One side is found by plugging in for A. Remember the perimeter of a square has the formula: P 4s We will simplify this with the answer: The perimeter is about inches. P 4s P 4( ) P Now we switch gears and do some graphing of inequalities. 4
5 Rules for graphing inequalities on a number line: 1) Use the initial rules for graphing points on a number line from Section ) 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. EXAMPLE 5: Graph each solution set. A. y > 7 B. p 3 / / 4 > so full circle not equal so open circle greater than so to right not equal so both directions HOMEWORK Page 123 #2159 odd 5
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