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Understanding Numbers in Algebra: Sets, Operations, and Properties

Explore different sets of numbers in algebra, from Natural Numbers to Real Numbers, including their properties and operations. Learn about rational, irrational, and whole numbers, as well as the fundamental properties of addition and multiplication in the algebraic field.

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Understanding Numbers in Algebra: Sets, Operations, and Properties

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  1. In Algebra we care about different sets of numbers and which numbers are part of different sets.

  2. Natural Numbers

  3. Natural Numbers•1, 2, 3, 4, 5, …

  4. Natural Numbers•1, 2, 3, 4, 5, …• Numbers we count with

  5. Natural Numbers•1, 2, 3, 4, 5, …• Numbers we count with•Positive whole numbers

  6. Natural Numbers•1, 2, 3, 4, 5, …• Numbers we count with•Positive whole numbers• Symbol = N

  7. Whole Numbers

  8. Whole Numbers•0, 1, 2, 3, 4, 5, …

  9. Whole Numbers•0, 1, 2, 3, 4, 5, …• Natural numbers & 0

  10. Whole Numbers•0, 1, 2, 3, 4, 5, …• Natural numbers & 0•Symbol = W

  11. Integers

  12. Integers• … , -3, -2, -1, 0, 1, 2, 3, …

  13. Integers• … , -3, -2, -1, 0, 1, 2, 3, …•Whole numbers and their opposites

  14. Integers• … , -3, -2, -1, 0, 1, 2, 3, …•Whole numbers and their opposites•Symbol = Z

  15. Rational Numbers

  16. Rational Numbers•Symbol = Q

  17. Rational Numbers•Symbol = Q• Numbers that can be written as the quotient of two integers

  18. Rational Numbers•Symbol = Q• Numbers that can be written as the quotient of two integers• “Normal” fractions

  19. Rational Numbers•For example … ¾ 5/3

  20. Rational Numbers•For example … ¾ 5/3 -½ 34/7

  21. Rational Numbers•For example … ¾ 5/3 -½ 34/7 2.25 -.66666…

  22. Rational Numbers•For example … ¾ 5/3 -½ 34/7 2.25 -.66666… 42 -11

  23. Rational Numbers•Symbol = Q• Numbers that can be written as the quotient of two integers• “Normal” fractions

  24. Irrational Numbers•Symbol = I or Ir• Not rational• Can’t be written as a quotient of integers• “Weird” numbers

  25. Examples of Irrational Numbers•Square roots you can’t simplify

  26. Examples of Irrational Numbers•Higher roots you can’t simplify

  27. Examples of Irrational Numbers•Special numbers  e 

  28. Examples of Irrational Numbers•Most trig function values sin(52) tan(107)

  29. Examples of Irrational Numbers•Decimals that don’t end and don’t repeat .27227722277722227777…

  30. Examples of Irrational Numbers 2

  31. Real Numbers•Symbol = R

  32. Real Numbers•Symbol = R• Rational and irrational numbers together

  33. Real Numbers•Symbol = R• Rational and irrational numbers together• Every number on the number line

  34. Real Numbers•Symbol = R• Rational and irrational numbers together• Every number on the number line• Every number you know

  35. Properties of Real Numbers• a.k.a. “Field Properties”

  36. A field is just any set that has the same properties as the real numbers.• The properties of numbersare essentially the postulates of algebra.

  37. Properties of Addition and Multiplication

  38. Commutative Property

  39. 3 + 5 = 5 + 32(-9) = -9  2Order doesn’t matter when you add of multiply.

  40. Commutative Property

  41. Associative Property

  42. -17 + (17 + 39) = (-17 + 17) + 39(7  4)  9 = 7(4  9)You can group together what you want to when you add of multiply.

  43. Associative Property

  44. Identity Property

  45. 7 + 0 = 4 0 + 2 = 25  1 = 5 1(-4) = -4When you add 0 or multiply by 1, you get back what you started with.

  46. Identity Property

  47. Inverse Property

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