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# Logic — the Art of Reasoning

Mathematics  — the Art of Studying Patterns Using Logic

# Algebra

#### By operational patterns I mean patterns that govern action. For example, People who have morining routines from the time they wake up till they go to work or school have operational patterns. Some get dressed before they brush their teeth others have breakfast before getting dressed and brushing their teet. In mathematics, an operational pattern is for example, the fact that when we multiply two numbers, say

5 x 9

#### we can break each number into two number that together add up to equal its value. In our example, we can do

(2 + 3 ) x ( 5 + 4 ).

#### Now if we want the product of the second multiplication to eqaul the product of the first, we can do

2 x 5 + 2 x 4 + 3 x 5 + 3 x 4 .

#### Experiencing this behavior of numbers for hundreds and thousands of years, led people to rule that

(a + b ) x ( c + d ) = a x c + a x d + b x c + b x d .

#### This is an operational pattern from mathematical viewpoint.

We can also look at this pattern as a consistent behavior of numbers. In this sense algebra deal with the patterns of the behavior of numbers.

Operational patterns are different from patterns of objects such as tiling. In math, patterns of objects are, for example, the even numbers, the sqaures of the integers and the prime numbers. While there are operational patterns to generate all the even number and all the sqaures of the integers, there is no operational pattern to generate all of the prime numbers.

## Interesting Algebraic Relationships

Here are a few interesting relationships that may challenge your imagination or provoke you to think in different ways.