A set of numbers can be divided into several subsets → integers, natural numbers, rational numbers, real numbers, or complex numbers. Today we will focus on the group of rational numbers - that is, numbers that can be written as a ratio of two natural numbers.

A fraction consists of three parts: the numerator, the fraction bar, and the denominator.

Note that a fraction only makes sense when the denominator is a number different from zero. Otherwise, the fraction is undefined.

## Simplifying Fractions

The fractions and represent the same number, i.e., `0.5`

. It is essentially irrelevant which form you use. However, it is customary to write fractions in their *simplest form*, which is a form that cannot be further simplified. The process of simplifying fractions is called reduction and is usually a relatively easy task. All you need to do is find a number `x`

by which both the numerator and denominator of the fraction can be divided without a remainder.

If we have the fraction , we can clearly see that both the numerator and denominator can be divided by two, giving us the fraction . However, with some fractions, it's not so straightforward. Try reducing the fraction . The process in this case is as follows: First, we need to find the greatest common divisor of the numbers `369`

and `15`

(see article Největší společný dělitel).

GCD(369, 15) = 3 The greatest common divisor is the number 3 → thus, we need to divide both the numerator and the denominator by three

Whenever you work with fractions, try to simplify them first. This will save you from dealing with unnecessarily large numbers.

## Expanding Fractions

Expanding fractions works on a similar principle as simplifying fractions - any fraction can be multiplied by any number `x`

(`x`

≠ 0) to get a fraction of the same value: .

This property of fractions is especially useful when trying to convert fractions to a common denominator.

## Adding Fractions

Fractions can only be added if the denominators of both fractions are the same. If they are not, we must expand one of the fractions by some number so that the denominators are the same.

Calculate: . The denominators of both fractions are the same → we can proceed with the addition. **Note: ** When adding fractions, we only add the numerators, and the denominator remains unchanged. Thus, the result is . We can further simplify the result by dividing by two:

Calculate: . The denominators are not the same → we need to find a common denominator. The common denominator is the number `12`

. Thus, we expand both fractions so that the denominator is `12`

: .

The general formula for adding fractions is:

## Subtracting Fractions

It is almost the same as adding fractions, except that the numerators are subtracted from each other:

## Multiplying Fractions

Multiplying fractions is probably one of the easiest operations you can perform with fractions. Simply multiply the numerator of the first fraction by the numerator of the second fraction to get the numerator of the resulting fraction. Apply the same procedure to get the denominator. In other words:

Multiply the numerators and denominators of both fractions:

Similarly, calculate the example `\frac{4}{3}\cdot\frac{1}{8}`

`\frac{4}{3}\cdot\frac{1}{8} = \frac{4\cdot 1}{3\cdot 8} = \frac{4}{24}` We can simplify the fraction:

In the previous example, we simplified after obtaining the result, but it is possible to simplify directly during the multiplication process. This simplification is called "cross-canceling" because we cancel the numerator of the first fraction with the denominator of the second fraction and the denominator of the first fraction with the numerator of the second fraction. Simplification follows the rules for reducing a normal fraction:

Cancel: We can further simplify by canceling the denominator of the first fraction with the numerator of the second fraction: Result =

The general formula for multiplying fractions is

## Dividing Fractions

If you already know how to multiply fractions, there is nothing to fear. Division is simply multiplying by the reciprocal. So if you get the example , you will certainly handle it.

We need to multiply the fraction by the reciprocal of the fraction The reciprocal is

**Don't be discouraged by finding the reciprocal of a fraction! It is nothing complicated, just swap the numerator and denominator.**

Simplify the following expression:

Start by simplifying the top and bottom parts of the largest fraction: Now remove the complex fraction and simultaneously multiply the expression by the reciprocal of Now multiply everything and simplify; the result is:

## Comparing Fractions

You can compare numbers, but can you do the same with fractions? Try comparing the fractions and . At first glance, it is not clear which fraction is larger. To determine this at a glance, you need to convert both fractions to the same denominator (in this case, it will be `9`

).

When fractions have the same denominator, you only compare the numerators.

## Solved Examples

**5) ** Upravte

**6)** Upravte výraz

Advanced manipulations of expressions can be found in article: Úpravy lomených výrazů