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

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

To learn more about symbolic representation of numbers, see Numeric to Symbolic Conversion.

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

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;

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.

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.