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1 –2: Properties of Real Numbers

1 –2: Properties of Real Numbers. {1, 2, 3, 4, 5, …}. Counting (Natural) Numbers. {0, 1, 2, 3, 4, 5, …}. Whole Numbers. {…–3, –2, –1, 0, 1, 2, 3 …}. Integers. All numbers that can be expressed as a/b, where both a and b are integers and b  0.

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1 –2: Properties of Real Numbers

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  1. 1–2: Properties of Real Numbers

  2. {1, 2, 3, 4, 5, …} Counting (Natural) Numbers

  3. {0, 1, 2, 3, 4, 5, …} Whole Numbers

  4. {…–3, –2, –1, 0, 1, 2, 3 …} Integers

  5. All numbers that can be expressed as a/b, where both a and b are integers and b 0. • Includescommon fractions, terminating decimals, repeating decimals, and integers. • They do not include non-repeating decimals, such as . Rational Numbers

  6. Irrational Numbers • Those numbers that cannot be expressed as a ratio of two integers • Includes non-terminating, non-repeatingdecimals and special numbers, such as πand

  7. Real numbers include all rational and irrational numbers. Real Numbers

  8. All Real Numbers Rational Numbers Irrational Numbers Integers Whole Numbers Counting Numbers

  9. All whole numbers are integers. • All integers are whole numbers. • All natural numbers are real numbers. • All irrational numbers are real numbers. Ponder the statements...True or False?

  10. Classify each of the following numbers using all the terms that apply: natural (counting), whole, integer, rational, irrational, and real. Classifying Numbers A) B) 3 C) D) –7

  11. Closure Property • Commutative Property • Associative Property • Identity Property • Inverse Property • Distributive Property • Properties of Equality Properties of Real Numbers

  12. When you combine any two numbers in a set, the answer is part of the set. • For example, when you add or multiply real numbers, the result is also a real number. a + b is a real number a x b is a real number • Learn more Closure Property

  13. Commutative means that the order does not make any difference. a + b = b + a a • b = b • a Examples 4 + 5 = 5 + 4 2 • 3 = 3 • 2 • The commutative property does not work for subtraction or division. Commutative Property

  14. Associative means that the grouping does not make any difference. (a + b) + c = a + (b + c) (ab) c = a (bc) Examples (1 + 2) + 3 = 1 + (2 + 3) (2 • 3) • 4 = 2 • (3 • 4) • The associative property does not work for subtraction or division. Associative Property

  15. Do not change the value! • Additive Identity – When you add zero to any number, the result is the same number a + 0 = a -6 + 0 = -6 • Multiplicative Identity – When you multiply a number by one, the result is the same number a • 1 = a 8 • 1 = 8 Identity Properties

  16. Inverse Properties • Undo an operation • Additive Inverse – when you add a number and its opposite, the result is 0 a + (-a) = 0 5 + (-5) = 0 • Multiplicative Inverse – when you multiply a number and its reciprocal, the result is 1

  17. Distributive property of multiplication with respect to either addition or subtraction. • a(b + c) = ab + bc • 3(4 - 7) = 3(4) - 3(7) • 3(2x + 4) = 3(2x) + 3(4) = 6x + 12 Distributive property

  18. Properties of Equality • Reflexive a = a • Symmetric If a = b, then b = a • Transitive If a = b and b = c, then a = c

  19. Real Numbers (mathisfun) • Properties of Real Numbers (regentsprep) • Math Properties (purplemath) • Properties of Equality (hotmath) • Glossary of Properties (dr.math/mathforum) More info…

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