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Transcript
Real Numbers and Properties Objective: The students will be able to classify real numbers and recognize different properties that exist with real numbers. M11.A.1.3.1 – Locate/identify irrational numbers at the approximate location on a number line. M11.A.1.3.2 – Compare and/or order any real numbers (rational and irrational may be mixed). Key Question • What is the difference between rational and irrational numbers and how else can numbers be classified? Classifying Numbers Set – a collection of objects • Real Numbers – all numbers are in this set (rational OR irrational) SUBSETS • Rational Numbers – can be written as a fraction or a decimal that stops or repeats Examples: ½ -7.34 0.11 • Irrational Numbers – decimals that never stop and never repeat Examples: 17 π 21 Are All Radicals Irrational? a Radicand – number under the radical No…If the radicand is a perfect square, the radical is rational. Perfect Squares Common Perfect Squares 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 • Whenever you take the square root (radical) of a perfect square, you get a number that is rational. The square root undoes squaring a number. 42 16 and 16 4 Classifying Numbers • Natural Numbers – “counting numbers” 1, 2, 3, 4, 5, … • Whole Numbers – includes 0 0, 1, 2, 3, 4, 5, … • Integers – positive or negative whole number or zero “dashes on the number line” {…-3, -2, -1, 0, 1, 2, 3, …} Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers Real Numbers Rational Integers Whole Natural Irrational Identify the sets into which each number belongs 1. -7 5. 22 2. 1/8 6. 0.91 3. π 7. 5 4. 0 8. 27 3 5+7=7+5 • Commutative Property of Addition: • Commutative Property of Multiplication: • Associative Property of Addition: • Associative Property of Multiplication: • Additive Identity Property: • Multiplicative Identity Property: • Additive Inverse Property: • Multiplicative Inverse Property: 4 3 1 3 4 • Zero Property of Multiplication: 30=0 32=23 5 + (4 + 2) = (5 + 4) + 2 2 (3 5) = (2 3) 5 5+0=5 81=8 12+ (-12) = 0 Tips to Help you Remember • Commutative Property: When you commute, you move from one place to another. In math, can move numbers for addition and multiplication. • Associative Property: When you associate with your peers, you are often in groups. In math, can group numbers in any order for addition and multiplication. • Identity Property: Your identity is who you are. In math, identity properties allow a number to get back to itself. • Inverse Property: The inverse properties look at opposites. Name that Property! 1. 5+0=5 2. 9+3=3+9 3. 0.8 + (-0.8) = 0 4. (4 + 2) + 1 = 4 + (2 + 1) Name that Property! 2 2 3 3 5. 1 6. 13(x + y) = 13x + 13y 7. 6 1 1 6 8. (ab)c = a(bc) 9. 9(2) = 2(9) Fill in the Blank 1. 15 + ___ = 12 + 15 2. 9(8 – 3) = (___ 8) – (___ 3) 3. 3 (4 2) = (3 4) ___ 4. 4 ___ = 1 5. 5 + ___ = 0