You can easily visualize a multiplication if you think of numbers as length and width and the product as area. Many of these complicated mathematical properties can be easily understood when we visualize them; something we are going to call ‘Visual Proof’. Here we will try to visually prove the theorem on sum of first n odd numbers.

The Theorem

The theorem states that the “sum of the first N odd natural numbers is always a perfect square and is equal to N^2”. 

This can be proved in a number of ways but, there is also a proof without words.

The Visual Proof


Start from the lower left corner. The first blue dot is 1. Then we have 3 green dots surrounding it. Next we have 5 blue dots. Then 7, 9, 11 and 13. As you can see, these dots are the series of first 7 odd numbers.

Try to visualize them in parts. When you see the group of first blue dot and the next 3 green dots, you find out that it is a 2 X 2 square. Now add the next 5 blue dots. You can see that it is a 3 X 3 square. Now add the next 7 green dots and it looks like a 4 X 4 square and so on.

Watch the animated Visual Proof in the video at top

Transcript of the video

Let us see, if, we can add, these numbers, in a way, that, they always form, a square.

One. It’s a square. 1,2,3. Again a square. 1,2,3,4,5. Again a square. 1,2,3,4,5,6,7. Again a square. And so on. 

Here is a, quick visualization, of the same thing. Hope you enjoyed, how, mathematics can be put in a way, that you can, see, what’s going inside the theorem.

More Fun Learning

This is a fun place to learn so if you love math jokes and riddles; you can find them here.

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