overview

Fractions are represented in various non-standard representations. $\frac{2}{3}$ and $\frac{5}{8}$ are two fractions given in two different place value $\frac{1}{3}$ and $\frac{1}{8}$ 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 $0,1,2,\cdots$ 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 $\text{received:}3$ or $\text{aligned:}3$.
$-3$ is represented as either $\text{given:}3$ or $\text{opposed:}3$.*

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 $3\frac{5}{8}$.

fractions are non-standard

To compare the two fractions $\frac{2}{6}$ and $\frac{1}{14}$, convert the fractions to like fractions $\frac{14}{42}$ and $\frac{3}{42}$ and compare the numerators $14$ and $3$.

To add the two fractions $\frac{2}{6}$ and $\frac{1}{14}$, convert the fractions to like fractions and add the numerators.

To multiply the two fractions $\frac{2}{6}$ and $\frac{1}{14}$, multiply the numerators and multiply the denominators $\frac{2\times 1}{6\times 14}$, which results in completely different form $\frac{2}{84}$.

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 $\frac{1}{2}$ is given as $\frac{5}{10}$.

Since the place-value or denominator is standardized, $\frac{5}{10}$ is represented as $0.5$, that is the denominator need not be mentioned. It is implicitly given.

Similarly $\frac{3}{4}$, which is equivalently $\frac{75}{100}$, is $0.75$ in decimal representation.

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

The fraction $\frac{3}{4}$, which is equivalently $\frac{75}{100}$, 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 $\frac{75}{100}$ which equals $\frac{7}{10}+\frac{5}{100}$. The decimal representation $0.75$ is understood as $\frac{7}{10}+\frac{5}{100}$.

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 $\textcolor[rgb]{}{r\frac{p}{q}=r+\frac{a}{10}+\frac{b}{100}+\frac{c}{1000}+\cdots}$ is given as $\textcolor[rgb]{}{r.abc\cdots}$ in decimal representation.
Example:
$32\frac{5}{8}=30+2+\frac{6}{10}+\frac{2}{100}+\frac{5}{1000}$
$=32.625$*

examples

What is the decimal representation of $\frac{3}{4}$?

The answer is "$0.75$".

$\frac{3}{4}$

$=\frac{75}{100}$

$=\frac{7}{10}+\frac{5}{100}$

$=0.75$

What is the decimal representation of $\frac{2}{25}$?

The answer is "$0.08$".

$\frac{2}{25}$

$=\frac{8}{100}$

$=\frac{0}{10}+\frac{8}{100}$

$=0.08$

What is the decimal representation of $\frac{1}{3}$?

Note that $\frac{1}{3}=\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}+\cdots$.

The answer is "$0.333\cdots$".

summary

Fractions are represented in various non-standard representations. $\frac{p}{q}$ and $\frac{l}{m}$ are two fractions given in two different place value $\frac{1}{q}$ and $\frac{1}{m}$ 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:
$\frac{3}{4}$

$=\frac{75}{100}$

$=\frac{7}{10}+\frac{5}{100}$

$=0.75$
$7$ has tenth ($\frac{1}{10}$) place value and $5$ has hundredth ($\frac{1}{100}$) 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__