Mark read that the ancient Greeks use to arrange pebbles to represent numbers. He used square on a grid instead of pebbles to model both triangular and square numbers
How can you divide a square number into q two triangular numbers?
(6 triangular number) and (9 sqaure)
Say represent 5² as a square:
OOOOO
OOOOO
OOOOO
OOOOO
OOOOO
and split it up into two triangles next to the diagonal:
O
OO
OOO
OOOO
OOOOO
and (it should be the mirror image)
OOOO
OOO
OO
O
So in general, for an n×n square, we can split it into the larger triangle of base n using n(n+1)/2 pebbles, and the smaller of side (n-1) using n(n-1)/2 pebbles.
The total:
n(n+1)/2+n(n-1)/2
=(n²+n +n²-n)/2
=n²
Thanks :)
You're welcome!
To divide a square number into two triangular numbers, you need to find two triangular numbers that, when added together, equal the square number. In this case, we have a square number of 9 (3^2) that we want to divide.
To find the triangular numbers that add up to 9, we need to express 9 as the sum of two triangular numbers. Triangular numbers are formed by arranging objects (in this case, squares on a grid) in the shape of an equilateral triangle.
To solve this, let's start by listing the triangular numbers up to 9:
1, 3, 6, 10, 15, 21, 28, 36, 45
We can see that the triangular numbers 6 and 3 each sum up to 9, so we can divide the square number 9 into 6 and 3 triangular numbers.
In terms of modeling this using squares on a grid, we could arrange 6 squares in the shape of a triangular number and 3 squares in the shape of another triangular number, ultimately forming a square of 9.
So, to divide a square number, such as 9, into two triangular numbers, you need to find two triangular numbers that add up to that square number.