overview

It was learned earlier that integers are directed whole numbers, with

• positive numbers representing "aligned" to a direction, and

• negative numbers representing "opposed" to a direction.

Fractions are part of a whole and with direction information included.

directed numbers

Whole numbers representation is not sufficient to represent directed numbers.

For example, consider the numbers in

• I received $3$ candies and

• I gave $3$ candies.

In the whole numbers, both these are represented as $3$.

In integers, the first is $+3$ and the second is $-3$.*
Integer numbers are represented as follows.
$3$ is represented as either $\text{received:}3$ or $\text{aligned:}3$.
$-3$ is represented as either $\text{given:}3$ or $\text{opposed:}3$.*

The English word "integer" means "untouched", that is, the numbers are not cut into smaller pieces or fractions.

Are fractions directed numbers?

Consider the problem that illustrates the answer. A girl receives $\frac{2}{3}$ candy from her brother. Later, she gives $\frac{1}{3}$ candy back.

These fractions have an additional property

• $\frac{2}{3}$ : amount *received*

• $\frac{1}{3}$ : amount *given*

In this problem, received and given represent two directions. And so, fractions are directed numbers.

**Fractions are Directed Numbers** : Fractions are represented as directed numbers with positive and negative signs.

eg: $\frac{2}{5}$ represents $\text{received:}\frac{2}{5}$ or $\text{aligned:}\frac{2}{5}$. This fraction represents $2$ counts of amount with $\frac{1}{5}$ place value.

$-\frac{2}{5}$ represents $\text{given:}\frac{2}{5}$ or $\text{opposed:}\frac{2}{5}$. This fraction represents $-2$ counts of amount with $\frac{1}{5}$ place value.

The word "directed" means aimed in a particular direction.

Absolute value of a directed number is given as the count or measure without the direction information.

The sign of a directed number is the direction information of the number.

The operator $\left|n\right|$ represents the absolute value of the number $n$.

examples

What is the absolute value of $-\frac{1}{3}$?

The answer is "$\frac{1}{3}$"

What is the absolute value of $\frac{1}{4}$?

The answer is "$\frac{1}{4}$"

What is $|-\frac{1}{2}|$?

The answer is "$\frac{1}{2}$"

What is the sign of $-\frac{1}{7}$

The answer is "negative"

What is $\left|\frac{2}{7}\right|$?

The answer is "$\frac{2}{7}$"

What is the sign of $\frac{2}{7}$?

The answer is "positive"

What is $|-\frac{3}{7}|$?

The answer is "$\frac{3}{7}$"

What is the sign of $-\frac{3}{7}$?

The answer is "negative"

summary

A fraction is either

• a positive fraction, for example $\frac{2}{5}$, or

• a negative fraction, for example $-\frac{2}{5}$

The absolute value of a fraction is the amount specified by the fraction without the direction information. eg: $|-\frac{2}{5}|=\frac{2}{5}$

The sign of a fraction is the direction information specified in the fraction. eg: sign of $-\frac{2}{5}$ is negative or $-$

Outline

The outline of material to learn "fractions" is as follows.

• * click here for detailed outline of Fractions *

→ __Part of whole__

→ __Dividing a group__

→ __Fractions as Directed numbers__

→ __Like and Unlike Fractions__

→ __Proper and Improper Fractions__

→ __Equivalent & Simplest form__

→ __Converting unlike and like Fractions__

→ __Simplest form of a Fraction__

→ __Comparing Fractions__

→ __Addition & Subtraction__

→ __Multiplication__

→ __Reciprocal__

→ __Division__

→ __Numerical Expressions with Fractions__

→ __PEMA / BOMA__