# Algebra of Objects

by Ron Kurtus (updated 18 January 2022)

In Algebra, an alphabetical letter can represent an object, number, or group of numbers. When the letter represents an object, algebraic rules are limited.

You can add, subtract, multiply, or divide an object by a number. You are limited in adding or subtracting different objects. You cannot multiply or divide different objects together.

Questions you may have include:

• What happens when dealing with a single object?
• What happens when dealing with several objects?
• What happens when multiplying or dividing objects?

This lesson will answer those questions.

## Dealing with one type of object

When you are considering one type of object, you can add the numbers together and subtract then (as long as the result is positive).

Let the letter "a" represent an apple. Then 2a represents two apples. Likewise, 2a + 3a represents two apples plus three apples. The sum equals five apples.

### Subtracting

You can also subtract apples: 6a − 4a = 2a.

But you cannot have a negative object. The result of 4a − 6a is not equal to −2a, since a negative apple is meaningless.

### Multiply or divide by number

You can multiply or divide an object by a number.

For example 5 pairs of apples would be: 5 * 2a = 10a.

You can also have a fraction of an apple: ½a is one half an apple.

## Dealing with different objects

If "a" represents an apple and "b" represents a banana, adding and subtracting is limited to the object type. But you can group the objects.

For example: 3a + 2b − 2a + b equals or results in a + 3b. You can only add or subtract the same type of object.

You can multiply or divide several objects by a number: 5(2a +3b) = 10a + 15b.

## Multiplying objects together

You cannot multiply or divide objects by each other. 2a * 3a is meaningless, since apple times apple has no real meaning.

## Summary

When an alphabetical letter represents an object, algebraic rules are limited. You can add, subtract, multiply, or divide an object by a number. You are limited in adding or subtracting different objects. You cannot multiply or divide different objects together.

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