Fractions : Fractions as Directed Numbers

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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 $received: 3$ $text(received:)3$ or $aligned: 3$ $text(aligned:)3$ .

$- 3$ $-3$ is represented as either $given: 3$ $text(given:)3$ or $opposed: 3$ $text(opposed:)3$ .

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 $\frac{2}{3}$ $2/3$ candy from her brother. Later, she gives $\frac{1}{3}$ $1/3$ candy back.

These fractions have an additional property

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

• $\frac{1}{3}$ $1/3$ : 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: $\frac{2}{5}$ $2/5$ represents $received: \frac{2}{5}$ $text(received:)2/5$ or $aligned: \frac{2}{5}$ $text(aligned:)2/5$ . This fraction represents $2$ $2$ counts of amount with $\frac{1}{5}$ $1/5$ place value.

$- \frac{2}{5}$ $-2/5$ represents $given: \frac{2}{5}$ $text(given:)2/5$ or $opposed: \frac{2}{5}$ $text(opposed:)2/5$ . This fraction represents $- 2$ $-2$ counts of amount with $\frac{1}{5}$ $1/5$ 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 $- \frac{1}{3}$ $-1/3$ ?
The answer is " $\frac{1}{3}$ $1/3$ "

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

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

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

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

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

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

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

summary

A fraction is either

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

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

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

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