Mathematical Fundamentals

Working with visuals instead of symbols can often make the difference between blindly memorizing an algebraic technique and understanding how and why that technique works.

For example, here's a visualization that shows why the sum of the first nn odd numbers is equal to n2:n^2:

Understanding Math Visually

                 

In the animation below, the specific cases are only shown up to the square number, 25.25. What is the next, specific sum in the pattern?

1+3+5+7+9+11=?1+3+5+7+9+11=?

Understanding Math Visually

                 

Now let's look at the difference of two specific square numbers: 9282.9^2 - 8^2.

What's left after the subtraction is just a unit-layer shell from the original square that can be broken into two, rectangular pieces.

Both rectangles have one side that is length 1.1. One of the rectangles is the full length of the larger square, 9,9, and the other is the length of the smaller square, 8.8.

So, 9282=9+8=17.9^2 - 8^2 = 9+8=17.

Understanding Math Visually

                 

This geometry problem can be solved with logic that extends from the algebra that you've just seen.

You have two squares, one larger than the other. If the larger square has integer side lengths and if the small square is one unit shorter on each side, the difference between the two areas will always be _______________.\text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}.

Understanding Math Visually

                 

What is the value of 202192?20^2-19^2?

You can drag the slider below the figure to change the size of the figure. Notice the size can't go up to 2020 so you'll need to think about how the pattern applies to larger squares.

Understanding Math Visually

                 

In a later chapter of this course, you'll see how this pattern extends when both of the squares can be any size.

Algebraically, if we refer to the side length of the larger square with the variable aa and the side length of the smaller square with the variable b,b, the difference between the two square values is

a2b2.a^2 - b^2.

The identity that you're going to be able to derive and understand is known as the "difference of squares" identity.

Understanding Math Visually

                 

See if you can use visual logic to solve this faster than you'd be able to do the arithmetic out by hand.

What is the value of 232172? 23^2 - 17^2?

Hint: The difference between the area of a 23×2323 \times 23 square and a 17×1717 \times 17 square can be split into two rectangles that each have one side of length 6.6.

Understanding Math Visually

                 

Here's the full identity. Also, below it, you can see what the visual derivation will look like. Even if you've already seen a purely algebraic derivation of the identity, this one might be easier to understand if you enjoy doing math that feels real and tactile!

The Difference of Squares Identity: a2b2=(a+b)(ab)a^2 - b^2 = (a+b)(a-b)

You can go either from a difference of two squares to a product of two differences, or just as easily, you can go the other way around, cutting a rectangle into two pieces that can reassemble into a square with a square "bite" taken out of it.

Understanding Math Visually

                 

Bonus Challenge

We can use the difference of squares identity to turn exponential calculations into quick multiplication, but it can also be used to turn a multiplication problem that might take you a full minute to do by hand into a more quickly calculable exponential form.

For example, without using a calculator, evaluate 43×37.43 \times 37.

This might help:

Hint: 40×40=160040 \times 40 = 1600

Understanding Math Visually

                 
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