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Thought-Process to Discover Knowledge

Welcome to nubtrek.

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.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsDecimalsFundamentals of Decimals

Standard form of Fractions: Decimals

Decimals are introduced as "standard form of fractions".

Fractions are part-of-whole, with different place-values specified as denominators. The Decimals are standard form of fractions with standard form for

 •  place-values : `1//10`, `1//100`, etc.

 •  position of digits : decimal point and order of digits after decimal point.



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In "number systems", we have learned the following.

 •  Whole numbers

 •  Integers

 •  Fractions

Let us revise these in few questions.

What are whole numbers?

  • whole numbers are used to count objects
  • whole numbers are `0,1,2,cdots`
  • both the above
  • both the above

The answer is "both the above".

What are integers?

  • integers are directed whole numbers
  • integers are directed whole numbers
  • integers are part of a whole

The answer is "integers are directed whole numbers".

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`.

What are fractions?

  • fractions are directed whole numbers
  • fractions are numbers representing part of a whole
  • fractions are numbers representing part of a whole

The answer is "fractions are numbers representing part of a whole".

Whole numbers and Integers representation is not sufficient to represent quantities of part of an object.

For example, A pizza is cut into `8` pieces.

`3` whole pizzas and `5` pieces of a cut pizza are remaining.

Whole numbers or integers represent them as two quantities: `3` pizzas and `5` pieces when one whole is cut into `8` pieces. This representation is descriptive.

The same in fractions is `3 5/8`.

Which of the following step helps to compare the two fractions `2/6` and `1/14`?

  • convert the fractions to like fractions and compare the numerators
  • convert the fractions to like fractions and compare the numerators
  • compare the numerators as the fractions are given

The answer is "convert the fractions to like fractions and compare the numerators".

Which of the following step helps to add the two fractions `2/6` and `1/14`?

  • convert the fractions to like fractions and add the numerators
  • convert the fractions to like fractions and add the numerators
  • add the numerators as the fractions are given

The answer is "convert the fractions to like fractions and add the numerators".

Which of the following step helps to multiply the two fractions `2/6` and `1/14`?

  • convert the fractions to like fractions and multiply the numerators
  • multiply the numerators and multiply the denominators
  • multiply the numerators and multiply the denominators

The answer is "multiply the numerators and multiply the denominators".

One observation in using fractions is that the numbers are of different place-values and require extra computational effort to do basic arithmetics like comparison, addition, subtraction, and multiplication.

 •  to compare, the fractions have to be converted to like-fractions

 •  to add or subtract, the fractions have to be converted to like-fractions

 •  to multiply, the numerator and denominators are multiplied separately, and the product is of different place-value to the multiplicand and multiplier.


What can be done to simplify this?

  • convert all the fractions to have standardized place-value form
  • convert all the fractions to have standardized place-value form
  • the fraction arithmetic cannot be simplified

The answer is "convert all the fractions to have standardized place-value form".

In whole numbers, we have chosen the place-value system as units, tens, hundreds, etc.

Extending the same, the place-value of decimals is chosen to be

 •  one tenth or `1//10`

 •  one hundredth or `1//100`

 •  one thousandth or `1//1000`

 •  etc.

By this, a fraction `1/2` is given as `5/10`.

Since the place-value or denominator is standardized, `5/10` is represented as `0.5`, that is the denominator need not be mentioned. It is implicitly given.

Similarly `3/4`, which is equivalently `75/100`, is `0.75` in decimal representation.

What is the place value of `5` in the decimal `0.5`?

  • one tenth
  • one tenth
  • there is no place value given for decimals

The answer is "one tenth".

The fraction `3/4`, which is equivalently `75/100`, is `0.75` in decimal representation.

What is the place value of `7` in `0.75`?

  • tenth
  • tenth
  • hundredth

The answer is "tenth".

The fraction `3/4`, which is equivalently `75/100`, is `0.75` in decimal representation.

What is the place value of `5` in `0.75`?

  • tenth
  • hundredth
  • hundredth

The answer is "hundredth".

Note that the number is given equivalently as `75/100` which is considered to be `7/10 + 5/100`. The decimal representation `0.75` is understood as `7/10 + 5/100`.

Decimal Representation : Decimal representation is the standard form of fractions. The place-value or denominator is standardized to power of `10`.

A mixed fraction `color(coral)(r p/q = r + a/10 + b/100 + c/1000 + cdots)` is given as `color(coral)(r.abc cdots)` in decimal representation.

Example:

`32 5/8 = 30 + 2 + 6/10 + 2/100 + 5/1000`

`= 32.625`

Solved Exercise Problem:

What is the decimal representation of `3/4`?

  • `0.25`
  • `0.75`
  • `0.75`

The answer is "`0.75`".

`3/4`
`=75/100`
`=7/10 + 5/100`
`=0.75`

Solved Exercise Problem:

What is the decimal representation of `2/25`?

  • `0.08`
  • `0.08`
  • `0.8`

The answer is "`0.08`".

`2/25`
`=8/100`
`=0/10 + 8/100`
`=0.08`

Solved Exercise Problem:

What is the decimal representation of `1/3`?

Note that `1/3 = 3/10 + 3/100 + 3/1000 + cdots`.

  • `1/3` cannot be represented in decimals
  • `0.333cdots`
  • `0.333cdots`

The answer is "`0.333cdots`".

                            
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