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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```

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

Create Symbolic Variables

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)`.

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. 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.