1. THE REAL NUMBER SYSTEM
Horan Math 8

2. Warm-Up Discussion Share with your group:
What is your favorite number and why?

3. Just think for a minute…
Where did numbers come from…
Math is DISCOVERED, not invented
Math is the language that nature speaks

4. Math IS the real world

5. Vocabulary for Number Theory
Natural Numbers – numbers used for counting
Positive, not a fraction or decimal
Example: 1, 27, 315
This may seem like a dumb question

6. Natural Numbers Who discovered natural numbers?
Basically like, everybody
The first numbers that humankind needed to survive
“I will trade you 1 elephant for 3 camels”
I will trade you 4 camels for 2 cows for a house. Final offer.”

7. Vocabulary for Number Theory
Natural Numbers – numbers used for counting
Positive, not a fraction or decimal
Example: 1, 27, 315
Whole Numbers – counting numbers AND zero
Zero or Positive, not a fraction or decimal
Example: 0, all natural numbers
This may seem like a dumb question

8. Whole Numbers Who discovered them? Egyptians (kinda)
Babylonians (kinda)
Mesoamericans (Mexico and Central America)
Mayans
Incans
Greeks (How can nothing be something?)
Chinese
Indians
But our zero came from Arabic culture around 600 AD

9. Pop Quiz True or False
Fifteen is a whole number

10. Pop Quiz - True or False Fifteen is a whole number TRUE!
All natural numbers are whole numbers

11. DON’T FORGET Whole numbers include ALL natural numbers!
They didn’t get rid of previous numbers when they discovered zero!

12. Vocabulary for Number Theory
Natural Numbers – numbers used for counting
Positive, not a fraction or decimal
Example: 1, 27, 315
Whole Numbers – counting numbers AND zero
Zero or positive, not a fraction or decimal
Example: 0, all natural numbers
Integers – positive or negative whole numbers
Not a fraction or a decimal
Example: 12, -4, 0, -978
This may seem like a dumb question

13. Integers Who discovered integers? Greeks around 200 B.C.?
Nope! They thought negatives were “false” answers.
Chinese in 100 A.D.?
Kinda! They used red and black rods to represent integers, but didn’t have a symbol to represent negative numbers
Indians in 700 A.D.?
Correct! Indians began using a “+” sign to represent negative numbers around 400 A.D. They used positive and negative integers to keep track of debt

14. Pop-Quiz - Integers
Negative fifteen is a natural number, but it is not an integer.

15. Pop-Quiz – True or False
Negative fifteen is a natural number, but it is not an integer.
FALSE! Negative fifteen is an integer, but not a natural number

16. How do negative numbers interact with each other?
Let’s practice…

17. Vocabulary for Number Theory
Natural Numbers – numbers used for counting
Positive, not a fraction or decimal
Example: 1, 27, 315
Whole Numbers – counting numbers AND zero
Zero or positive, not a fraction or decimal
Example: 0, all natural numbers
Integers – positive or negative whole numbers
Not a fraction or a decimal
Example: 12, -4, 0, -978
This may seem like a dumb question

18. Vocabulary for Number Theory (Cont’d)
Rational Numbers – a number that can be expressed as a fraction or a decimal that terminates/repeats
Any integer, decimal or fraction
Example: ½ , , 4, , 0

19. Rational Numbers Who discovered them?
Egyptian Hieroglyphs show a purely decimal system in existence around 3000 B.C.
That means people formalized the concept of fractions and decimals before they thought of zero!

20. Warm-up– What labels would you use?
( )
Is it a…
Natural number?
Whole number?
Integer?
Rational Number?

21. Pop Quiz – What labels would you use?
( ) = 0
Is it a…
Natural number?
Whole number?
Integer?
Rational Number?

22. Pop Quiz – What labels would you use?
( ) = 0
It is a…
Natural number?
Whole number.
Integer.
Rational Number.

23. DON’T FORGET!
Simplify before you try to categorize a number

24. Roots and Exponents 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100

25. Roots and Exponents
Square rooting is the opposite of squaring

26. Perfect Roots √1 = 1 √4 = 2 √9 = 3 √16 = 4 √25 = 5 √36 = 6 √49 = 7
√64 = 8
√81 = 9
√100= 10

27. A Square Root can be simplified…
If it is 1, 4, 9, 16, 25, 49, 64…

28. Vocabulary for Number Theory (Cont’d)
Rational Numbers – a number that can be expressed as a fraction or a decimal that terminates/repeats
Any integer, decimal or fraction
Example: ½ , , 4, , 0
Irrational Numbers – a number that, when written as a decimal, goes on forever without repeating…
Non-Perfect Square Roots, Pi (π), Phi (ϕ) or e
Example: √7, 2π, √15, π

29. Irrational Numbers Who discovered them?
Pythagoras, the Greek Mathematician
Used a relationship between right triangles to discover irrational numbers
(More on him later)

30. Pop Quiz! – Find the Irrational Numbers
√16 9π
-17 2
1.45 36
√14

31. Pop Quiz! – Find the Irrational Numbers
√16 9π
-17 2
1.45 36
√14

32. Vocabulary for Number Theory (Cont’d)
Rational Numbers – a number that can be expressed as a fraction or a decimal that terminates/repeats
Any integer, decimal or fraction
Example: ½ , , 4, , 0
Irrational Numbers – a number that, when written as a decimal, goes on forever without repeating…
Non-Perfect Square Roots, Pi (π), Phi (ϕ) or e
Example: √7, 2π, √15, π
Real Numbers – all numbers that can be put on a number line
All numbers we will discuss in this class are real
Example: ANY NUMBER

33. Pop Quiz – TRUE OR FALSE 21 -710 + (½ ⋅ 11) – (13 ÷ 5)
The result of this problem will be a real number

34. Pop Quiz – TRUE OR FALSE 21 -710 + (½ ⋅ 11) – (13 ÷ 5)
The result of this problem will be a real number
TRUE! All numbers we will discuss in class are real numbers

35. Vocab list (so far…) Natural Number Whole Number Integer
Rational Number
Irrational Number
Real Number