A **fraction** is a number expressed as the quotient of two mathematical expressions, often integers. Fractions come in the form $ \frac{a}{b} $ , where $ a $ is called the numerator and $ b $ is called the denominator.

Sometimes the word "fraction" means the quotient of two integers. Under this definition, $ \frac34 $ is a fraction, but $ \frac1{\pi+\sqrt2} $ is not.

## Types of fractions

If the numerator is smaller than the denominator, it is called a proper fraction, and is less than one. An example of this is $ \frac35 $ .

If the numerator is larger than the denominator, it is called an **improper fraction**, and is greater than one. An example of an improper fraction is $ \frac{51}{2} $ , which is 25 wholes and a half;
^{[1]}
$ 25\frac12 $ as a mixed number.

In addition, $ \frac{a}{b}=a\div b $ . For example, $ 2\frac63 $ would be $ 2\times (6\div3) $ .

## Manipulating fractions

Multiplying fractions is very easy:

- $ \frac{a}{b}\cdot\frac{c}{d}=\frac{a\cdot c}{b\cdot d} $

For $ a\ne0 $ , $ \frac{a}{a}=1 $ . This gives us a useful identity:

- $ \frac{a}{b}=\frac{a}{b}\cdot\frac{c}{c}=\frac{a\cdot c}{b\cdot c} $

allowing us to "force" values of the numerator or denominator. Some call this *multiplying by one creatively*.

Adding fractions is somewhat harder. If both the summands have the same denominator, we can use the distributive property:

- $ \frac{a}{b}+\frac{c}{b}=a\cdot\frac1b+c\cdot\frac1b=(a+c)\cdot\frac1b=\frac{a+c}{b} $

But if they're different, we can multiply by one creatively to force their denominators to line up:

- $ \frac{a}{b}+\frac{c}{d}=\frac{ad}{bd}+\frac{bc}{bd}=\frac{ad+bc}{bd} $

This formula is not terribly useful to memorize; knowing the technique is much more helpful. If we're working with integers, instead of $ bd $ in the denominator, we can use the least common multiple of $ b,d $.

### Example

- $ \frac29+\frac56=\frac{12}{54}+\frac{45}{54}=\frac{57}{54}=\frac{19}{18} $

Note that we could have taken a shortcut using the least common multiple:

- $ \frac29+\frac56=\frac4{18}+\frac{15}{18}=\frac{19}{18} $

We multiplied the first fraction by 2/2 and the second by 3/3.

## Conversions, mixed numbers and improper fractions

*Explain in more detail what mixed numbers and improper fractions are.*

You can convert a **mixed number** (a, surprisingly, mixed fraction — $ 5\frac45 $ is a mixed number) into an improper fraction (a fraction in which the numerator is larger than the denominator) by doing three things:

- Multiply the whole number (the $ 5 $) by the denominator (the other $ 5 $) of the fractional part. In the example above, to change $ 5\frac45 $ to a mixed number, first multiply $ 5\times5=25 $ .
- Add the numerator of the fractional part to that product. In the example, $ 4+25=29 $ .
- The resulting sum is the numerator of the new (improper) fraction, with the denominator remaining the same in both the mixed number and the improper fraction. In the example, $ 5\frac45=\frac{29}{5} $ .

Likewise, you can convert an improper fraction into a mixed number by following another 3 rules:

- Divide the numerator by the denominator. Let's say, for example, $ \frac{32}{7} $. $ 32\div7=4 $ with 4 left over (the
*remainder*) - The quotient (without the remainder) becomes the whole number part of the mixed number. The remainder becomes the numerator of the fractional part. In the example, 4 is the whole number part and 4 is the numerator of the fractional part.
- The new denominator is the same as the denominator of the improper fraction. In the example, the denominator is 7. Thus $ \frac{32}{7}=4\frac47 $ .

## Rationalizing the denominator

In elementary algebra, we are sometimes asked to **rationalize the denominator** of an expression like this one:

- $ \frac{1+\sqrt5}{\sqrt{11}-\sqrt7} $

To rationalize the denominator is to create a fraction equal to this one with an integer in the denominator.

As with adding fractions, we multiply by one creatively. The trick here is to use the difference of squares factorization $ (a-b)(a+b)=a^2-b^2 $ . We can turn the $ \sqrt{11}-\sqrt7 $ into a whole number by multiplying it by $ \sqrt{11}+\sqrt7 $ :

- $ (\sqrt{11}-\sqrt7)(\sqrt{11}+\sqrt7)=\sqrt{11}^2-\sqrt7^2=11-7=4 $

So by multiplying our fraction by $ \frac{\sqrt{11}+\sqrt7}{\sqrt{11}+\sqrt7} $ , we get a nice, round 4 in the denominator:

- $ \frac{1+\sqrt5}{\sqrt{11}-\sqrt7}=\frac{1+\sqrt5}{\sqrt{11}-\sqrt7}\cdot\frac{\sqrt{11}+\sqrt7}{\sqrt{11}+\sqrt7}=\frac{(1+\sqrt5)(\sqrt{11}+\sqrt7)}{(\sqrt{11}-\sqrt7)\cdot(\sqrt{11}+\sqrt7)}=\frac{\sqrt{11}+\sqrt7+\sqrt{55}+\sqrt{35}}{4} $

## Fractal fractions

Fractal fractions are fractions on which the numerator and/or the denominator consist of infinite other fractions. A simple fraction or number can be expressed as an infinite fraction in this way:

- 1. Express a number as a fraction with its numerator or denominator including the number. All square roots have this property, for example:
- $ \sqrt2=\frac2{\sqrt2} $
- 2. Now, replace the part of the fraction with the chosen number with the fraction itself. This step has to be repeated infinitely on the new fractions:
- $ \sqrt2=\frac2{\frac2{\sqrt2}} $

- $ \sqrt2=\frac2{\frac2{\frac2{\frac2{\ldots}}}} $

Then, the fractions can be split into a sum of two different fractions.

### Fractal fraction simplification

To convert an infinite fraction to a simpler one or a number equations must be used since taking a part of the infinite part of the fraction doesn't lead to the result it corresponds to:

- $ \frac22=1\ne\sqrt2 $
- $ \frac2{\frac22}=\frac21=2\ne\sqrt2 $
- The equation in this example is:
- $ x=\frac2{\frac2{\frac2{\ldots}}}\rArr x=\frac2x\rArr x^2=2\rArr x=\sqrt2 $

## References

- ↑ Wikipedia:Fraction (mathematics)#Mixed numbers.

- Divide the numerator by the denominator. In the example, $ \tfrac{11}{4} $ , divide 11 by 4. $ 11\div 4=2 $ with remainder 3.
- The quotient (without the remainder) becomes the whole number part of the mixed number. The remainder becomes the numerator of the fractional part. In the example, 2 is the whole number part and 3 is the numerator of the fractional part.
- The new denominator is the same as the denominator of the improper fraction. In the example, they are both 4. Thus $ \tfrac{11}{4}=2\tfrac34 $"