1. REAL NUMBERS AND THE NUMBER LINE Math 230 Presentation By Sigrid Robiso

2. Rational Numbers The ancient Greeks created a reasonable method of measuring What is a rational number? Is every number a rational number? Euclid

3. Pythagorean Theorem Pythagoras Remember the Pythagorean Theorem: a² + b² = c²

4. Pythagorean Theorem a² + b² = c² example if a= 1 and b= 1 1² + 1² = c² c= √ 2 which is an irrational number because it is not equal to the ratio of two numbers

5. The Real Number Line

6. Any rational number corresponds to any point on this line For ex. 5/2 on the line

7. But how do we find an irrational point on the number line?

8. Irrational Numbers on the Number Line 1. Build a square whose base is the interval from 0 to 1

9. Irrational Numbers on the Number Line 2. Next draw the diagonal from 0 to the upper right corner of the square 3. Using a compass copy the length of that diagonal line onto the number line and make a mark

10. Irrational Numbers on the Number Line Remember: √ 2 is irrational which makes us question is there a uniform method to label every point on the line- rational and irrational?

11. The Decimal Point Let’s consider the decimal expansion of √ 2 √ 2 = 1.414213562…

12. √ 2= 1.414213562 The number left to the decimal point shows that our number will be somewhere between 1 and 2 Where? We cut the interval from 1 to 2 into 10 equal pieces The next digit, 4, tells us in which small interval our number is located. We then take that small interval and cut it up into 10 very small equal pieces The next digit, 1, tells us which very small interval our number resides

13. The Decimal Point √ 2= 1.414213562…. Notice: as we continue this process we break the intervals smaller and smaller This process allows us to pin point our number √ 2 We keep getting closer to the √ 2 but this process never ends for √ 2 because it is irrational We keep pin pointing into smaller and smaller intervals but we must repeat this process infinitely many times to pinpoint the placement of √ 2 exactly