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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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Directed Numbers

This page introduces integers as directed numbers. The positive numbers are aligned to the chosen direction and the negative numbers are opposed to the chosen direction.

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number 5 whole The number of flowers in the picture is 5.

This is an example of using a whole number to count or measure quantities.

Some early humans naturally used hands to count similar to the numbers `1, 2, 3, cdots`. Natural numbers or Counting numbers are `1, 2, 3, cdots`

Whole numbers are `0` along with natural numbers.

Let us recall what is a whole number.

A whole number is a count or measurement of a quantity.

 •  Number of pencils is `20`.

 •  Length of a pencil is `14`cm

 •  Mass of a iron ball is `2`kg

 •  Volume of a bottle of water is `1` liter

 •  Time taken to finish a work is `30`mins

 •  Temperature of water is `26^@`C

 •  Price of a pen is `15` coins

Consider a girl and her brother sharing candies. She receives `2` candies from her brother. The number of candies received is `2`.

On another day, The girl gives `2` candies to her brother. The number of candies given is `2`.

A girl received `2` candies from her brother. This is represented by the number `2`.

On another day, she gives `2` candies to her brother. This is represented by the number `2`.

Note that both the numbers are represented as `2`. Are these two quantities same?

No, 2 received is different from 2 given.

Consider the two situations.

A girl receives `2` candies from her father. On another day, she receives another `2` candies from her father. Effectively, she has received, `2+2=4` candies from her father.

The girl receives `2` candies from her brother. On another day, she gives `2` candies to her brother. Overall, she has received `2` candies and returned `2` candies. This makes total of, `2-2=0` candies received from her brother.

From the example of a girl receiving `2` candies and giving `2` candies, it is noted that the `2` received and `2` given are different.

`2` received is aligned to the direction of receiving. And `2` given is opposed to the direction of receiving. They together make `0`.

A girl receives `10` candies from her brother. Later, she gives `2` candies to her brother.

She has `10-2=8` candies.

It is noted that the given is subtracted.

The two inherent properties of the numbers

 •  The quantities received and given represent two different values.

 •  When combined, the quantities received and given are subtracted.

In accordance with these, negative numbers are introduced. Minus sign is adopted to provide the missing information in the numbers.

For Example:

`2` received is `+2` or `2`

`2` given is `-2`

The count of candies is understood in two different forms

 •  `5` candies received.

 •  `5` candies given.

Similarly other quantities have two different forms.

 •  Height above and below ground level

 •  Distance to reach a city and distance covered after crossing the city

 •  Length or Mass or Volume : amount available to use and amount used in excess

 •  Temperature above `0^@`C or below `0^@`C

 •  Temperature to heat or cool down

 •  Time remaining or time passed since start

 •  Money credited or debited

All these applications are modeled with numbers given in the form
`3` and `-3`
which are equivalently given as `text(received:)3 = 3` and `text(given:)3 = -3` to explain some fundamentals in arithmetics.

Such numbers with positive and negative direction are called Directed Numbers.

Integers are directed whole numbers.

The word "negative" means: absence of something; not having or does not exist.

familiarize with the terminology

The word "positive" means: presence of something; having or existing.

familiarize with the terminology

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

The positive numbers represent numbers aligned to a direction and negative numbers represent numbers opposed to the direction.

familiarize with the terminology

Integers : Directed whole numbers are called integers.

Solved Exercise Problem:

If `2` degree above `0` is `+2`, what is `7` degree below `0`?

  • `+7`
  • `-7`
  • `-7`

The answer is '`-7`'.

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