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# Algebraic Operations and Symbols

by Ron Kurtus (revised 18 January 2022)

Common * algebraic operations* are addition, subtraction, multiplication, and division. Special operations include raising to a power and taking roots. Algebraic operations work the same way as arithmetic operations.

These operations have their own * symbols*. There are also special symbols for equals, greater than, and less then.

You can perform operations on constants, variables, terms, or expressions.

Questions you may have include:

- What are common algebraic operations and their symbols?
- What are some special algebraic operations?
- What are the equality symbols?

This lesson will answer those questions.

## Common operations

Common mathematical operations indicate one entity operating on another. Typical symbols used are:

- Addition
**+** - Subtraction
**−** - Multiplication
**×** - Division
**÷**

Multiplication and division symbols have some variations.

### Variations with multiplication sign

In Arithmetic, **×** is used as the multiplication sign. However, in Algebra, the letter "**x**" is often used as a variable or unknown, so using **×** as the multiplication sign can cause confusion.

One solution is to use an asterisk (*****) or a dot (**·**) to indicate multiplication. Instead of **x × y**, multiplication can be designated as **x*y** or **x·y**.

But another problem then pops up. If you use the dot when dealing with numbers, it may be confused with a decimal point. For example, writing **2** times **5 ** as **2·5** might be confused with **2½** or **2.5**. With numbers, it is better to use *****, as in **2*5**, or even the **×** sign.

### Using no multiplication sign

When multiplying variables, constants, or a number times a variable, you often see the multiplication sign completely omitted. Instead of writing **x** times **y** times **z** as **x*y*z**, it is usually written as **xyz** in Algebra. This is a common practice as a way to make things less cluttered.

#### Numbers first

When including numbers in the multiplication, the number is written first and no multiplication sign is used. Thus, **x** times **3** is written as **3x**, and **x** times **2** times **a** is written as **2ax**.

#### Multiplying numbers together

But if you are multiplying two or more numbers together, you **must include** a multiplication sign. **2** times **5** times **3** is **not** written as **253** but as **2*5*3**.

If there is a letter involved, try to make it as clear as possible. **2*5x** or **2*5*x** are both acceptable.

### Division sign

The slash (**/**) is also used to denote division. Thus, **a ÷ b** and **a/b** both mean "**a** divided by **b**."

## Special algebraic operations

Special operations include raising to a power, taking the square root, summation, and integral.

### Raising to a power

If you wanted to multiply a number by itself several times, you could write **7 * 7 * 7 * 7** or **a * a * a**.

A shorthand way of doing that is called "raising to a power" and consists of writing the number, constant, or variable and placing a small number to the upper right hand corner, which represents how many times the multiplication was repeated.

7 * 7 * 7 * 7would then be7and is called^{4}7 to the 4th power.

a * a * acan be written asaand is called^{3}a to the third poweror commonly calleda-cubed.

ais called^{2}a-squared.

Sometimes the symbol "**^**" is used to indicate raising to a power, especially on a computer. For example:

7^4can be used instead of7.^{4}

a^3can be used instead ofa^{3}

### Taking a root

Taking the root of a number is somewhat abstract. For example, the square root of a number is the number that when multiplied by itself equals that number. For example, the square root of **9** is **3**, since **3 * 3 = 9**.

The symbol for square root is "**√**" or **SQRT**. For example:

√(25) = 5orSQRT(25) = 5

### Parentheses

Parentheses is a way of grouping items, such that an operation acts on everything within the parentheses.

√(20 + 5)means the square root operation acts on25.

## Equality symbols

Equality symbols include equals, greater than, and less than.

### Equals

When one variable is the same or equal to another, you can use the equals symbol: **"=**"

For example **a = 6** indicates that the variable **a** is equal to the number **6**.

### Greater than

The *greater than* symbol is "**>**". Sometimes it is indicated as **GT**. For example **7 > 6** and **a GT b**.

If one quality is greater than or equal to the other, you can use the **GTE** symbol.

### Less than

Likewise, the less than symbol is "**<**" or **LT**. For example **6 < 7**.

Also *less than or equal* to is indicated by **LTE**.

## Summary

Common algebraic operations are addition, subtraction, multiplication, and division. Special operations include raising to a power and taking roots. These operations have their own symbols. There are also special symbols for equals, greater than, and less then.

You can perform operations on constants, variables, terms, or expressions.

Always do your best

## Resources and references

### Websites

**Algebraic operation** - Wikipedia

### Books

(Notice: The *School for Champions* may earn commissions from book purchases)

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## Algebraic Operations and Symbols