THE REAL NUMBER SYSTEM Horan Math 8

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

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

Math IS the real world https://www.youtube.com/watch?v=HpI9sk7TgNE

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

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.”

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

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

Pop Quiz - True or False Fifteen is a whole number

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

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

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

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

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

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

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

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

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 , -2.75, 4, -15, 0

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!

Warm-up– What labels would you use? (-10 + 10) Is it a… Natural number? Whole number? Integer? Rational Number?

Pop Quiz – What labels would you use? (-10 + 10) = 0 Is it a… Natural number? Whole number? Integer? Rational Number?

Pop Quiz – What labels would you use? (-10 + 10) = 0 It is a… Natural number? Whole number. Integer. Rational Number.

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

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

Roots and Exponents Square rooting is the opposite of squaring

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

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

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 , -2.75, 4, -15, 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, 3 + 5 π

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

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

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

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 , -2.75, 4, -15, 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, 3 + 5 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

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

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

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