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1-5 Roots and Real Numbers Warm Up Simplify each expression. 2. 112 121 1. 62 36 3. (–9)(–9) 81 25 36 4. Write each fraction as a decimal. 5. 2 0.4 6. 5 0.5 5 9 7. 5 3 5.375 8 Holt McDougal Algebra 1 8. –1 5 6 –1.83 1-5 Roots and Real Numbers Objectives Classify numbers within the real number system. Holt McDougal Algebra 1 1-5 Roots and Real Numbers Real numbers can be classified according to their characteristics. Natural numbers are the counting numbers: 1, 2, 3, … Whole numbers are the natural numbers and zero: 0, 1, 2, 3, … Integers are the whole numbers and their opposites: –3, –2, –1, 0, 1, 2, 3, … Holt McDougal Algebra 1 1-5 Roots and Real Numbers Rational numbers are numbers that can be expressed in the form , where a and b are both integers and b ≠ 0. When expressed as a decimal, a rational number is either a terminating decimal or a repeating decimal. • A terminating decimal has a finite number of digits after the decimal point (for example, 1.25, 2.75, and 4.0). • A repeating decimal has a block of one or more digits after the decimal point that repeat continuously (where all digits are not zeros). Holt McDougal Algebra 1 1-5 Roots and Real Numbers Irrational numbers are all numbers that are not rational. They cannot be expressed in the form where a and b are both integers and b ≠ 0. They are neither terminating decimals nor repeating decimals. For example: 0.10100100010000100000… After the decimal point, this number contains 1 followed by one 0, and then 1 followed by two 0’s, and then 1 followed by three 0’s, and so on. This decimal neither terminates nor repeats, so it is an irrational number. Holt McDougal Algebra 1 1-5 Roots and Real Numbers If a whole number is not a perfect square, then its square root is irrational. For example, 2 is not a perfect square and is irrational. Holt McDougal Algebra 1 1-5 Roots and Real Numbers The real numbers are made up of all rational and irrational numbers. Holt McDougal Algebra 1 1-5 Roots and Real Numbers Reading Math Note the symbols for the sets of numbers. R: real numbers Q: rational numbers Z: integers W: whole numbers N: natural numbers Holt McDougal Algebra 1 1-5 Roots and Real Numbers Let’s Try together… Write all classifications that apply to each real number. A. –32 32 1 –32 = –32.0 –32 = – –32 can be written in the form . –32 can be written as a terminating decimal. rational number, integer, terminating decimal B. irrational Holt McDougal Algebra 1 14 is not a perfect square, so irrational. is 1-5 Roots and Real Numbers Let’s Try together… Write all classifications that apply to each real number. a. 7 7 4 can be written in the form 9 . can be written as a repeating decimal. rational number, repeating decimal b. –12 –12 can be written in the form . –12 can be written as a terminating decimal. rational number, terminating decimal, integer 67 9 = 7.444… = 7.4 Holt McDougal Algebra 1 1-5 Roots and Real Numbers Let’s Try together… Write all classifications that apply to each real number. irrational 10 is not a perfect square, so is irrational. 100 is a perfect square, so is rational. 10 can be written in the form and as a terminating decimal. natural, rational, terminating decimal, whole, integer Holt McDougal Algebra 1 1-5 Roots and Real Numbers Find each square root. 1. 3 2. 3. 5 4. 1 5. The area of a square piece of cloth is 68 in2. Estimate to the nearest tenth the side length of the cloth. 8.2 in. Write all classifications that apply to each real number. 6. –3.89 rational, repeating decimal Holt McDougal Algebra 1 7. irrational 1-5 Roots and Real Numbers Try each problem WITHOUT looking at your notes Write all possible classifications of each number 1. -53 integer Natural, whole, integer, 2. rational, terminating decimal 3. 2/3 Holt McDougal Algebra 1 Rational, repeating decimal 1-5 Roots and Real Numbers 4. 5. 1/2 √12 6. –3.89 7. Holt McDougal Algebra 1 Rational, terminating decimal irrational Rational, repeating decimal irrational 1-5 Roots and Real Numbers Always, Sometimes, Never… 1.A negative number is an integer 2.A repeating decimal is irrational 3.Mixed numbers are rational 4.A terminating decimal is a rational number Holt McDougal Algebra 1 1-5 Roots and Real Numbers Always, Sometimes, Never… 1.A negative number is an integer SOMETIMES -√2 2. A repeating decimal is irrational NEVER 3. Mixed numbers are rational ALWAYS 4. A terminating decimal is a rational number ALWAYS Holt McDougal Algebra 1 1-5 Roots and Real Numbers Pre-Requisite Packet – Front page… Holt McDougal Algebra 1