maths > fractions

Fractions as Directed Numbers

what you'll learn...

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$$3$ candies and
• I gave $3$$3$ candies.

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

In integers, the first is $+3$$+ 3$ and the second is $-3$$- 3$.

Integer numbers are represented as follows.

$3$$3$ is represented as either $\text{received:}3$$\textrm{\left(r e c e i v e d\right\rangle} 3$ or \text{aligned:}3$\textrm{\left(a l i g \ne d\right\rangle} 3$.

$-3$$- 3$ is represented as either $\text{given:}3$$\textrm{\left(g i v e n\right\rangle} 3$ or $\text{opposed:}3$$\textrm{\left(o p p o s e d\right\rangle} 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}$$\frac{2}{3}$ candy from her brother. Later, she gives $\frac{1}{3}$$\frac{1}{3}$ candy back.

These fractions have an additional property

•  $\frac{2}{3}$$\frac{2}{3}$ : amount received

•  $\frac{1}{3}$$\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}$$\frac{2}{5}$ represents $\text{received:}\frac{2}{5}$$\textrm{\left(r e c e i v e d\right\rangle} \frac{2}{5}$ or \text{aligned:}\frac{2}{5}$\textrm{\left(a l i g \ne d\right\rangle} \frac{2}{5}$. This fraction represents $2$$2$ counts of amount with $\frac{1}{5}$$\frac{1}{5}$ place value.

$-\frac{2}{5}$$- \frac{2}{5}$ represents $\text{given:}\frac{2}{5}$$\textrm{\left(g i v e n\right\rangle} \frac{2}{5}$ or $\text{opposed:}\frac{2}{5}$$\textrm{\left(o p p o s e d\right\rangle} \frac{2}{5}$. This fraction represents $-2$$- 2$ counts of amount with $\frac{1}{5}$$\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 $|n|$$| n |$ represents the absolute value of the number $n$$n$.

examples

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

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

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

What is the sign of $-\frac{1}{7}$$- \frac{1}{7}$
The answer is "negative"

What is $|\frac{2}{7}|$$| \frac{2}{7} |$?
The answer is "$\frac{2}{7}$$\frac{2}{7}$"

What is the sign of $\frac{2}{7}$$\frac{2}{7}$?
The answer is "positive"

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

What is the sign of $-\frac{3}{7}$$- \frac{3}{7}$?
The answer is "negative"

summary

A fraction is either

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

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

Outline