Math 409/409G History of Mathematics

1. Math 409/409GHistory of Mathematics Figurative Numbers

2. What’s a figurative number? The figurative numbers are the number of dots that can be evenly placed on the sides of successively nested polygons of the same shape. For example:

3. Triangular numbers 1, 3, 6, 10, …

4. Square numbers 1, 4, 9, 16, …

5. Pentagonal numbers 1, 5, 12, 22, …

6. Triangular Numbers The nth triangular number is denoted by tn.

7. Let’s look for a pattern. In general the sum of the first n counting numbers.

8. But what is this sum?

9. And multiplying both sides of this formula by 2 gives us the number theory formula for the sum of the first 2n even counting numbers, namely:

10. Square Numbers The nth square number is denoted by sn. Since we’re dealing with square arrays, it’s obvious that in general

11. Let’s look for a pattern. In general the sum of the first 2n - 1 odd counting numbers.

12. Since sn= n2, we get the formula

13. In the beginning of this lesson you saw an algebraic proof that Let’s look at a geometric proof of this.

14. s4= = = - So , and in general,

15. This ends the lesson on Figurative Numbers