# Numeric to Symbolic Conversion

This topic shows how Symbolic Math Toolbox™ converts numbers into symbolic form. For an overview of symbolic and numeric arithmetic, see Choose Numeric or Symbolic Arithmetic.

To convert numeric input to symbolic form, use the `sym` command. By default, `sym` returns a rational approximation of a numeric expression.

```t = 0.1; sym(t)```
```ans =  $\frac{1}{10}$```

`sym` determines that the double-precision value `0.1` approximates the exact symbolic value $\frac{1}{10}$. In general, `sym` tries to correct the round-off error in floating-point inputs to return the exact symbolic form. Specifically, `sym` corrects round-off error in numeric inputs that match the forms $\frac{\mathit{p}}{\mathit{q}}$, $\frac{\mathit{p}\pi }{\mathit{q}}$, ${\left(\frac{\mathit{p}}{\mathit{q}}\right)}^{\frac{1}{2}}$, ${2}^{\mathit{q}}$, and ${10}^{\mathit{q}}$, where $\mathit{p}$ and $\mathit{q}$ are modest-sized integers.

For these forms, demonstrate that `sym` converts floating-point inputs to the exact symbolic form. First, numerically approximate $\frac{1}{7}$, $\pi$, and $\frac{1}{\sqrt{2}}$.

`N1 = 1/7`
```N1 = 0.1429 ```
`N2 = pi`
```N2 = 3.1416 ```
`N3 = 1/sqrt(2)`
```N3 = 0.7071 ```

Convert the numeric approximations to exact symbolic form. `sym` corrects the round-off error.

`S1 = sym(N1)`
```S1 =  $\frac{1}{7}$```
`S2 = sym(N2)`
`S2 = $\pi$`
`S3 = sym(N3)`
```S3 =  $\frac{\sqrt{2}}{2}$```

You can force `sym` to accept the input as is by placing the input in quotes. Demonstrate this behavior on the previous input `0.142857142857143`. The `sym` function does not convert the input to `1/7`.

`sym('0.142857142857143')`
`ans = $0.142857142857143$`

When you convert large numbers, use quotes to exactly represent them. Demonstrate this behavior by comparing `sym(133333333333333333333)` with `sym('133333333333333333333')`.

`sym(1333333333333333333)`
`ans = $1333333333333333248$`
`sym('1333333333333333333')`
`ans = $1333333333333333333$`

You can specify the technique used by `sym` to convert floating-point numbers using the optional second argument, which can be `'f'`, `'r'`, `'e'`, or `'d'`. The default flag is `'r'`, for rational form.

### Conversion to Rational Symbolic Form

Convert input to exact rational form by calling `sym` with the `'r'` flag. This is the default behavior when you call `sym` without flags.

```t = 0.1; sym(t,'r')```
```ans =  $\frac{1}{10}$```

### Conversion by Using Floating-Point Expansion

If you call `sym` with the flag `'f'`, `sym` converts double-precision, floating-point numbers to their numeric value by using $\mathit{N}\cdot {2}^{\mathit{e}}$, where $\mathit{N}$ and $\mathit{e}$ are the exponent and mantissa respectively.

Convert `t` by using a floating-point expansion.

`sym(t,'f')`
```ans =  $\frac{3602879701896397}{36028797018963968}$```

### Conversion to Rational Symbolic Form with Error Term

If you call `sym` with the flag `'e'`, `sym` returns the rational form of `t` plus the error between the estimated, exact value for `t` and its floating-point representation. This error is expressed in terms of `eps` (the floating-point relative precision).

Convert `t` to symbolic form. Return the error between its estimated symbolic form and its floating-point value.

`sym(t,'e')`
```ans =  $\frac{\mathrm{eps}}{40}+\frac{1}{10}$```

The error term `eps/40` is the difference between `sym('0.1')` and `sym(0.1)`.

### Conversion to Decimal Form

If you call `sym` with the flag `'d'`, `sym` returns the decimal expansion of the input. The `digits` function specifies the number of significant digits used. The default value of `digits` is 32.

`sym(t,'d')`
`ans = $0.10000000000000000555111512312578$`

Change the number of significant digits by using `digits`.

```digitsOld = digits(7); sym(t,'d')```
`ans = $0.1$`

For further calculations, restore the old value of `digits`.

`digits(digitsOld)`

### Conversion to Variable Precision

You can create symbolic numbers with variable-precision floating-point arithmetic by using `vpa`. By default, `vpa` calculates values to 32 significant digits.

`piVpa = vpa(pi)`
`piVpa = $3.1415926535897932384626433832795$`

When you use `vpa` on a numeric input, such as `log(2)`, the numeric expression is first evaluated to the MATLAB® default double-precision number that has less than 32 significant digits. Then, `vpa` is applied on that double-precision number, which can be less accurate. For more accurate results, convert numeric expressions to symbolic expressions with `sym` and then use `vpa` to evaluate the results with variable precision. For example, find `log(2)` with 17 and 20 digits precision.

`vpaOnDouble = vpa(log(2))`
`vpaOnDouble = $0.69314718055994528622676398299518$`
`vpaOnSym_17 = vpa(log(sym(2)),17)`
`vpaOnSym_17 = $0.69314718055994531$`
`vpaOnSym_20 = vpa(log(sym(2)),20)`
`vpaOnSym_20 = $0.69314718055994530942$`

When you convert large numbers, use quotes to exactly represent them.

`inaccurateNum = vpa(123456789012345678)`
`inaccurateNum = $123456789012345680.0$`
`accurateNum = vpa('123456789012345678')`
`accurateNum = $123456789012345678.0$`