### 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 $n$ odd numbers is equal to $n^2:$

# Understanding Math Visually

In the animation below, the specific cases are only shown up to the square number, $25.$ Using the trick shown in the animation, what is the sum of $1+3+5+7+9+11=?$

# Understanding Math Visually

Now let's look at the difference of two specific square numbers: $9^2 - 8^2.$

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 $1.$ One of the rectangles is the full length of the larger square, $9,$ and the other is the length of the smaller square, $8.$

So, $9^2 - 8^2 = 9+8=17.$

# 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 $20^2-19^2?$

You can drag the slider below the figure to change the size of the figure. Notice the size can't go up to $20$ 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 $a$ and the side length of the smaller square with the variable $b,$ the difference between the two square values is

$a^2 - b^2.$

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 $23^2 - 17^2?$

Hint: The difference between the area of a $23 \times 23$ square and a $17 \times 17$ square can be split into two rectangles that each have one side of length $6.$

# 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: $a^2 - b^2 = (a+b)(a-b)$

You can go either from a difference of two squares to a product of difference and a sum, 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 $43 \times 37.$

This might help:

Hint: $40 \times 40 = 1600$

# Understanding Math Visually

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