## Create Symbolic Numbers, Variables, and Expressions

This page shows how to create symbolic numbers, variables, and expressions. To learn how to work with symbolic math, see Perform Symbolic Computations.

### Create Symbolic Numbers

You can create symbolic numbers by using `sym`. Symbolic numbers are exact representations, unlike floating-point numbers.

Create a symbolic number by using `sym` and compare it to the same floating-point number.

```sym(1/3) 1/3```
```ans = 1/3 ans = 0.3333```

The symbolic number is represented in exact rational form, while the floating-point number is a decimal approximation. The symbolic result is not indented, while the standard MATLAB® result is indented.

Calculations on symbolic numbers are exact. Demonstrate this exactness by finding `sin(pi)` symbolically and numerically. The symbolic result is exact, while the numeric result is an approximation.

```sin(sym(pi)) sin(pi)```
```ans = 0 ans = 1.2246e-16```

### Create Symbolic Variables

You can create symbolic variables using either `syms` or `sym`. Typical uses of these functions include:

• `sym` – Create numbered symbolic variables or create symbolic variables in MATLAB functions.

• `syms` – Create fresh symbolic variables for interactive symbolic workflows, that is, for symbolic variable creation at the MATLAB command line or in MATLAB live scripts. A fresh symbolic variable does not have any assumptions.

The `syms` command is shorthand for the `sym` syntax, but the two functions handle assumptions differently. For more details, see Reuse Names of Symbolic Objects.

Create the symbolic variables `x` and `y` using `syms` and `sym`, respectively.

```syms x y = sym('y')```

The first command creates a symbolic variable `x` in the MATLAB workspace with the value `x` assigned to the variable `x`. The second command creates a symbolic variable `y` with the value `y`.

With `syms`, you can create multiple variables in one command. Create the variables `a`, `b`, and `c`.

`syms a b c`

If you want to create a MATLAB array of numbered symbolic variables, the `syms` syntax is inconvenient. Therefore, use `sym` instead to create an array of many numbered symbolic variables.

Clear the workspace. Create a row vector containing the symbolic variables `a1, ..., a20` and assign it to the MATLAB variable `A`. Display the variable in the MATLAB workspace.

```clear all A = sym('a', [1 20]) whos```
```A = [ a1, a2, a3, a4, a5, a6, a7, a8, a9, a10,... a11, a12, a13, a14, a15, a16, a17, a18, a19, a20] Name Size Bytes Class Attributes A 1x20 8 sym ```

`A` is a `1`-by-`20` array of 20 symbolic variables.

By combining `sym` and `syms`, you can create many fresh symbolic variables with corresponding variables name in the MATLAB workspace.

Clear the workspace. Create the fresh symbolic variables ```a1, ..., a10``` and assign them the MATLAB variable names `a1, ..., a10`, respectively. Display the variables in the MATLAB workspace.

```clear all syms(sym('a', [1 10])) whos```
``` Name Size Bytes Class Attributes a1 1x1 8 sym a10 1x1 8 sym a2 1x1 8 sym a3 1x1 8 sym a4 1x1 8 sym a5 1x1 8 sym a6 1x1 8 sym a7 1x1 8 sym a8 1x1 8 sym a9 1x1 8 sym ```

The MATLAB workspace contains 10 MATLAB variables that are symbolic variables.

The `syms` command is a convenient shorthand for the `sym` syntax, and its typical use is to create fresh symbolic variables for interactive symbolic workflows. Use the `sym` syntax to create the following:

• Symbolic variables in MATLAB functions

• Many numbered symbolic variables

• Symbolic variable whose value differs from its name in the MATLAB workspace

• Symbolic number, such as `sym(5)`

• Symbolic variable that inherits the assumptions from a previously used symbolic variable having the same name

### Create Symbolic Expressions

Suppose you want to use a symbolic variable to represent the golden ratio

`$\phi =\frac{1+\sqrt{5}}{2}$`

The command

`phi = (1 + sqrt(sym(5)))/2;`

achieves this goal. Now you can perform various mathematical operations on `phi`. For example,

`f = phi^2 - phi - 1`

returns

```f = (5^(1/2)/2 + 1/2)^2 - 5^(1/2)/2 - 3/2```

Now suppose you want to study the quadratic function `f` = `ax`2 + `bx` + `c`. First, create the symbolic variables `a`, `b`, `c`, and `x`:

`syms a b c x`

Then, assign the expression to `f`:

`f = a*x^2 + b*x + c;`

Tip

To create a symbolic number, use the `sym` command. Do not use the `syms` function to create a symbolic expression that is a constant. For example, to create the expression whose value is `5`, enter `f = sym(5)`. The command `f = 5` does not define `f` as a symbolic expression.

### Reuse Names of Symbolic Objects

If you set a variable equal to a symbolic expression, and then apply the `syms` command to the variable, MATLAB software removes the previously defined expression from the variable. For example,

```syms a b f = a + b```

returns

```f = a + b```

If later you enter

```syms f f```

then MATLAB removes the value `a + b` from the expression `f`:

```f = f```

You can use the `syms` command to clear variables of definitions that you previously assigned to them in your MATLAB session. `syms` clears the assumptions of the variables: complex, real, integer, and positive. These assumptions are stored separately from the symbolic object. However, recreating a variable using `sym` does not clear its assumptions. For more information, see Delete Symbolic Objects and Their Assumptions.

## Support

#### Mathematical Modeling with Symbolic Math Toolbox

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