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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 use two ways to create symbolic variables, `syms`

and `sym`

.
The `syms`

syntax is a shorthand for `sym`

.

Create 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 value `y`

.
Therefore, the commands are equivalent.

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 many variables, the `syms`

syntax
is inconvenient. Instead of using `syms`

, use `sym`

to
create many numbered variables.

Create the variables `a1, ..., a20`

.

A = sym('a', [1 20])

A = [ a1, a2, a3, a4, a5, a6, a7, a8, a9, a10,... a11, a12, a13, a14, a15, a16, a17, a18, a19, a20]

The `syms`

command is a convenient shorthand
for the `sym`

syntax. Use the `sym`

syntax
when you create many variables, when the variable value differs from
the variable name, or when you create a symbolic number, such as `sym(5)`

.

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.
However, `syms`

does not clear the following assumptions
of the variables: complex, real, integer, and positive. These assumptions
are stored separately from the symbolic object. For more information,
see Delete Symbolic Objects and Their Assumptions.

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