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Standard form of Fractions: Decimals


    what you'll learn...

overview

Fractions are represented in various non-standard representations. 23 and 58 are two fractions given in two different place value 13 and 18 respectively.

The place-value of decimals is standardized to be

 •  one tenth or 1/10

 •  one hundredth or 1/100

 •  one thousandth or 1/1000

 •  etc.

warmup to learn

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

 •  Whole numbers

 •  Integers

 •  Fractions

Let us revise these.




Whole numbers are 0,1,2, and are used to count or measure.

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 received:3 or aligned:3.

-3 is represented as either given:3 or opposed:3.


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 358.

fractions are non-standard

To compare the two fractions 26 and 114, convert the fractions to like fractions 1442 and 342 and compare the numerators 14 and 3.


To add the two fractions 26 and 114, convert the fractions to like fractions and add the numerators.


To multiply the two fractions 26 and 114, multiply the numerators and multiply the denominators 2×16×14, which results in completely different form 284.


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.


To simplify this, we can convert all the fractions to have standardized place-value form. That is, all fractions can be expressed with the same denominator. Is it possible to standarize this?

standardize

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 12 is given as 510.

Since the place-value or denominator is standardized, 510 is represented as 0.5, that is the denominator need not be mentioned. It is implicitly given.

Similarly 34, which is equivalently 75100, is 0.75 in decimal representation.


the place value of 5 in the decimal 0.5 is one tenth.

The fraction 34, which is equivalently 75100, is 0.75 in decimal representation.

The place value of 7 in 0.75 is "tenth".

The place value of 5 in 0.75 is "hundredth"


Note that the number is given equivalently as 75100 which equals 710+5100. The decimal representation 0.75 is understood as 710+5100.

standardized & non-standarsized

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

A mixed fraction rpq=r+a10+b100+c1000+ is given as r.abc in decimal representation.

Example:

3258=30+2+610+2100+51000

=32.625

examples

What is the decimal representation of 34?
The answer is "0.75".

34
=75100
=710+5100
=0.75


What is the decimal representation of 225?
The answer is "0.08".

225
=8100
=010+8100
=0.08


What is the decimal representation of 13?

Note that 13=310+3100+31000+.
The answer is "0.333".

summary

Fractions are represented in various non-standard representations. pq and lm are two fractions given in two different place value 1q and 1m respectively.

The place-value of decimals is standardized to be

 •  one tenth or 1/10

 •  one hundredth or 1/100

 •  one thousandth or 1/1000

 •  etc. eg: 34
=75100
=710+5100
=0.75 7 has tenth (110) place value and 5 has hundredth (1100) place value.

Outline

The outline of material to learn "decimals" is as follows.

Note: goto detailed outline of Decimals

•   Decimals - Introduction

    →   Decimals as Standard form of Fractions

    →   Expanded form of Decimals

•   Decimals - Conversion

    →   Conversion between decimals and fractions

    →   Repeating decimals

    →   Irrational Numbers

•   Decimals - Arithmetics

    →   Comparing decimals

    →   Addition & Subtraction

    →   Multiplication

    →   Division

•   Decimals - Expressions

    →   Expression Simplification

    →   PEMA / BOMA