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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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mathsFractionsFractions : Fundamentals

Fractions as Directed Numbers

Fractions are also directed numbers. That is, Fractions with positive and negative values are explained.



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Whole numbers representation is not sufficient to represent directed numbers.

For example, consider the numbers in
• I received `3` candies and
& bull; 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. A girl receives `2/3` candy from her brother. Later, she gives `1/3` candy back.

These fractions have an additional property

 •  `2/3` : amount received

 •  `1/3` : amount given

In this problem, received and given represent two directions.

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

eg: `2/5` represents `text(received:)2/5` or `text(aligned:)2/5`. This fraction represents `2` counts of amount with `1/5` place value.

`-2/5` represents `text(given:)2/5` or `text(opposed:)2/5`. This fraction represents `-2` counts of amount with `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 `| |` represents the absolute value of the number.

Solved Exercise Problem:

What is the absolute value of `-1/3`?

  • `1/3`
  • `1/3`
  • `0`

The answer is "`1/3`"

Solved Exercise Problem:

What is the absolute value of `1/4`?

  • same value `1/4`
  • same value `1/4`
  • `-1/4`

The answer is "`1/4`"

Solved Exercise Problem:

What is `|-1/2|`?

  • positive number `1/2`
  • positive number `1/2`
  • `-1/2`

The answer is "`1/2`"

Solved Exercise Problem:

What is the sign of `-1/7`

  • `1/7`
  • negative
  • negative

The answer is "negative"

Solved Exercise Problem:

What is `|2/7|`?

  • positive fraction `2/7`
  • positive fraction `2/7`
  • `-27`

The answer is "`2/7`"

Solved Exercise Problem:

What is the sign of `2/7`?

  • positive
  • positive
  • negative

The answer is "positive"

Solved Exercise Problem:

What is `|-3/7|`?

  • positive fraction `3/7`
  • positive fraction `3/7`
  • `-3/7`

The answer is "`3/7`"

Solved Exercise Problem:

What is the sign of `-3/7 `?

  • positive
  • negative
  • negative

The answer is "negative"

                            
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