# Secrets of Algebra

The last article had a nice response (thanks for that). So today something from the world of "forgotten math" - have fun!

Arithmetic can often not prove some of its strongholds by vague means. In these cases we need more general algebra methods. For these types of arithmetic theorems, which are algebraically justified, there are many rules for abbreviated arithmetic operations. **Speed multiplication**:

In the old days without computers or calculators, great arithmeticists used many simple algebraic tricks; to make your life easier:*The "x" is representative of multiplication (we were too lazy to try LaTeX :-))*

Let's look at:

988² =?

Can you solve it in your head?

It's very simple, let's take a closer look:

988 x 988 = (988 + 12) x (998 -12) + 12² = 1000 x 976 + 144 = 976 144

It's also easy to understand what's going on here:

(a + b) (a - b) + b² = a² - b² + b² = a²

OK so far so good. Now let's try to do the math quickly - even combinations like

986 x 997, without calculator!

986 x 997 = (986 - 3) x 1000 + 3 x 14 = 983 042

What happened here? We can write down the factors as follows:

(1000-14) x (1000-3)

1000x1000 - 1000x14 - 1000x3 + 14x3

Let's play with the factors:

1000 (1000 - 14) - 1000 x 3 + 14 x 3 =

1000x986 - 1000x3 + 14x3 =

1000 (986 - 3) + 14 x 3

That's all!

Let's study another powerful technique of algebra that can be used to compute some mathematical operations in our head based on:

a² = (a + b) x (a-b) + b²

Examples:

27² = (27 + 3) x (27-3) + 3 = 30 x 24 + 9 = 729

63² = 66 x 60 + 3 = 3

54² = 58 x 50 + 4 = 2

It's most fun when the last number is 5:

35²: 3 x 4 = 12; 5² = 25 = 1

65²; 6 x 7 = 42; 5² = 25 = 4

Mathematics can be so beautiful!