*Glenda Boozer*

# Mathematical Symbols: List & Meanings

## Math Is a Language of Symbols

Imagine that you have a pet hamster that hasn't stopped growing since the day you got it. To track and analyze its amazing growth, you'll need to use a range of mathematical symbols. In math, **symbols** are concise ways to show mathematical relations and operations.

In this lesson, we'll stick with the symbols that you'll see most often. Remember, if all else fails, you can always write things out in words, but make sure to ask someone about the correct symbol later.

## Relation Symbols

To get a true idea of your giant hamster's scale, you'll need to show how its size relates to other quantities. Relation symbols tell us how one **expression**, a mathematical representation of a quantity, relates to another expression. You probably know the equal sign: =. It tells us that the expressions on both sides of it are exactly equal to one another. So, you could say your hamster's weight equals 205 pounds.

While that gives an exact value for its weight, you could use this sign, which means that the expressions are only approximately equal:

For example, your hamster may be approximately equal in size to a baby elephant.

When things are not equal, one is greater than the other, so we use greater than and less than signs.

- >
- <

These show which expression is less: each symbol points to the lesser expression.

These symbols, which are similar to the greater than and less than signs, are the greater than or equal to and less than or equal to signs:

These signs include the amount itself, whereas greater than or less than refer to numbers right up to, but not including, that exact amount. Think of the difference between your hamster being at least 200 pounds and greater than 200 pounds.

## Basic Arithmetic

As your hamster continues its alarming growth, you'll need to know basic arithmetic symbols. We know that + means addition, and - means subtraction, and you can use these symbols to show how its size has changed.

But, what if you wanted to convert its weight from pounds to another unit of measurement? In that case, you'd use multiplication or division. However, we can't always use this common multiplication symbol because it may sometimes be confused with the variable *x*:

Instead we use the asterisk: *

Or we can use a dot:

Similarly, we can either use this symbol to represent division: ÷ or a slash /.

Along with these goes this symbol that stands for either adding or subtracting the value that comes after it: ±. If you noticed that your hamsters eats about 40 pounds of cabbage a day, but sometimes he eats 35 or 45 pounds, then you could say that he eats 40 ± 5 pounds a day.

## Grouping Symbols

As your hamster calculations become increasingly complex, you'll need grouping symbols to help you figure out the order of operations.

- Parentheses look like this: ( ).
- Square brackets look like this:
- Curly brackets look like this: { }

All of them are used to group items together. Grouping symbols are used to tell you to do the math of the items in the group first. The curly and square brackets are generally used to group items that are already in parentheses. Grouping symbols may also be used to indicate multiplication.

Parentheses also have some specialized meanings. When we write f(*x*), this does not mean f times *x*. It means the value of the function f when the input value is *x*. Parentheses are also used to show the coordinates of a point on a graph, like this: (3,2). When we write 3,2 we are making a short list with the numbers 3 and 2.

## Powers and Roots

Hopefully, your hamster isn't experiencing exponential growth. Just in case, we can raise an expression to any power by using the exponent symbol, which looks like this:

The small number is the exponent. This example means *x*-squared.

Roots are the opposite of powers: 3 is the square root of 9 because 3^2 = 9, -2 is the cube root of -8 because (-2)^3 = -8 and 2 is the fifth root of 32 because 2^5 = 32. The square root image looks like this:

And the cube root looks like this:

We can also have fourth roots, fifth roots and so on.

## Special Numbers

It might be that your hamster never stops growing. Hopefully that's not the case, but even if it is, we have a symbol for an infinitely large number. We call it infinity:

There's also a number with an infinite number of digits, which we get by dividing the circumference of any circle by its diameter. We call it pi:

## Miscellaneous Symbols

Just to make sure you have all the tools you need to keep track of your hamster's size, here are some symbols that don't necessarily fit into the previous categories.

The absolute value shows the distance of a number from zero on the number line, as in |5| = 5 and |-5| = 5.

The factorial symbol is even easier. In math, we use an exclamation point to denote factorials (!). This just means multiplying that number by every integer that comes before it until you reach 1.

- 5! = 5 * 4 * 3 * 2 * 1

On the other hand, something like this can be quite difficult to type:

So, we write it out as the summation from *x* = 1 to n of f(*x*).

It means f(1) + f(2) + f(3) + . . . + f(n).

Of course, the percent sign symbol means hundredths.

## Symbols Used in Calculus

Say you want to do more than just measure your hamster's weight every day. You want to measure its change over time and predict what that change will look like. This is where calculus and its symbols can be of use.

Delta stands for how much a variable has changed or the difference between two measurements. This is what the symbol looks like:

In calculus, we also have symbols called limits.

For example, we'd write this out as the limit, as *x* approaches a, of f(*x*).

We can write derivatives, which also help us look at rates of change, as f '(*x*), f ''(*x*), f'''(*x*) or f^n(*x*) for the first, second, third and nth derivatives, respectively, or like this:

Integrals are a fundamental component of calculus. Definite integrals that look like this can be written out as words: the integral from a to b of f(*x*) with respect to *x*:

Indefinite integrals like this one are similarly written out: the integral of f(*x*) with respect to *x*:

## Lesson Summary

Whether or not you own a giant hamster, we need to recognize and understand math symbols so that we can understand what other people are writing, and so we are able to write the symbols ourselves. We have symbols for relations like <, >, = and so on for the basic operations. We have grouping symbols that can work together (or, in the case of parentheses, that have some special uses). We have symbols for special values like pi and infinity, for roots and powers, other special operations and for calculus notation.

## Key Terms

- symbols: signs or images used as concise ways to show mathematical relations and operations
- expression: a mathematical representation of a quantity relating to another expression

## Learning Outcome

After reviewing this lesson, you should able to identify and describe the purpose of various mathematical symbols.

To unlock this lesson you must be a Study.com Member.

Create your account

### Register to view this lesson

### Unlock Your Education

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.

Become a MemberAlready a member? Log In

Back