Chapters

## Definition of a Fraction

Fractions are numbers that look like where because division by is meaningless.

is the top of the fraction and is called the numerator.

is the bottom of the fraction and is called the denominator.

#### Examples

All of these numbers are fractions

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## Division: Fractions as Quotients

A fraction is also known as a quotient of to . A quotient of to tells us to split the number into parts and it is equivalent to division.

We may also say that is broken into parts or that we divide the number by .

#### Example

Split into parts or perform the division of

split into parts is equivalent to saying or where

## Rational Numbers: Ratio of Integers

Many fractions are also Rational numbers. The Rational Numbers are a special case of fractions where is any integer and is any integer besides . The Integers are the positive and negative whole numbers along with , like on a number line.

The Rational in Rational number means that there is a ratio between the two numbers and . The big stands for quotient. The big for the Integers is from the German word for numbers 'zahlen'.

There are many fractions that are not rational, not the ratio of 2 integers. These numbers are called irrational numbers and have a decimal value that is never-ending (infinite) and non-repeating.

We will be sticking to fractions that are Rational numbers throughout these lessons.

## Negative Fractions

If a fraction is negative, we put the negative sign in the numerator or out in front of the fraction itself

If the negative sign happens to be in the denominator, don't write .

## Fractional Units

A fractional unit is 'one part numerator' of the 'whole'.

The 'whole' is the number that we 'split the number into', which is the denominator.

#### Example

The green shaded box represents 'one part' of the entire thing, which is 'split' into parts.

The 'whole' thing would be .

#### Example

The green shaded box represents unit out of total units that make up the 'whole' thing.

There are of these fractional units that make up the object.

## Fractional Values

#### Example

The green boxes represent of the whole amount or parts out of .

#### Example

The green boxes represent of the whole amount or parts out of .

## Fractions of a Number

We can take fractional amounts of numbers other than . We can look at it as a multiplication exercise, a division exercise or a mixture of both.

What is of ?

Multiplication

Division

What is of ?

or

## Fractions: Ratios and Proportions

When comparing different amounts or types of the same thing, you can form fractions from the ratios or proportions.

#### Example

Suppose we have crayons: blue  red  green  yellow.

The crayons can be placed in a ratio of blue : red : green : yellow.

We can form 4 fractions: 1 for the amount that each color of crayon is of the whole amount of crayons

are blue

are red

are green

are yellow

## Proper Fractions

Proper fractions have the values of their numerators less than the value of their denominators.

They have a value greater than and less than if positive

They have a value less than and greater than if negative

## Improper Fractions

Improper fractions are fractions that have their numerators larger than their denominators, which means they have a value greater than 1.

## Mixed Numbers

A mixed number is the union of an integer and a fractional amount less than .

They are equivalent to their improper fractional form.

#### Example

Express as a mixed number.

Divide the numerator by the denominator to obtain the highest whole number amount that will divide into it evenly.

Here it is because .

The remainder is then the numerator of the fraction less than that we attach next to the whole number, .

## Decimal Fractions

Decimal fractions have a power of in the denominator.

## Equivalent Fractions

2 fractions and are equivalent if and only if the product of the extremes equals the product of the means

and are known as the extremes, while and are known as the means

#### Example

Determine whether is equivalent to

and

and

## Simplifying Fractions

Most often we want fractions to be in simplest form. This means that the fraction is in lowest terms.

Lowest terms means that the numerator and denominator have no common factors or are relatively prime.

#### Example

and are not in lowest terms but are all equal to .

Both the numerators and denominators of each fraction have a common factor.

In , both the and the have as a common factor.

In , both the and the have as a common factor.

In , both the and the have as a common factor.

The simplified form of each of these fractions is .

#### Example

Is in lowest terms?

In , both and have the common factor of at least .

Actually, they both have the factor in common

We first factored out a to make and then factored out another to make and .

#### Example

Is in lowest terms?

We took the Prime Factorization of both and and cancelled out the common factors of and .

## Irreducible Fractions

Irreducible fractions are those that cannot be simplified any further (the numerator and denominator are in lowest terms). The numerator and denominator do not have any common factors and are called co-prime.

#### Examples

All of these fractions are irreducible or in lowest terms

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