maths > fractions

Fractions as Directed Numbers

    what you'll learn...


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 received:3 or aligned:3.

-3 is represented as either given:3 or 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 23 candy from her brother. Later, she gives 13 candy back.

These fractions have an additional property

 •  23 : amount received

 •  13 : 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: 25 represents received:25 or aligned:25. This fraction represents 2 counts of amount with 15 place value.

-25 represents given:25 or opposed:25. This fraction represents -2 counts of amount with 15 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| represents the absolute value of the number n.


What is the absolute value of -13?
The answer is "13"

What is the absolute value of 14?
The answer is "14"

What is |-12|?
The answer is "12"

What is the sign of -17
The answer is "negative"

What is |27|?
The answer is "27"

What is the sign of 27?
The answer is "positive"

What is |-37|?
The answer is "37"

What is the sign of -37?
The answer is "negative"


A fraction is either

 •  a positive fraction, for example 25, or

 •  a negative fraction, for example -25

The absolute value of a fraction is the amount specified by the fraction without the direction information. eg: |-25|=25

The sign of a fraction is the direction information specified in the fraction. eg: sign of -25 is negative or -


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