 maths > decimals

Standard form of Fractions: Decimals

what you'll learn...

overview

Fractions are represented in various non-standard representations. $\frac{2}{3}$$\frac{2}{3}$ and $\frac{5}{8}$$\frac{5}{8}$ are two fractions given in two different place value $\frac{1}{3}$$\frac{1}{3}$ and $\frac{1}{8}$$\frac{1}{8}$ respectively.

The place-value of decimals is standardized to be

•  one tenth or $1/10$$1 / 10$

•  one hundredth or $1/100$$1 / 100$

•  one thousandth or $1/1000$$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$$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$$3$ candies and
& bull; I gave $3$$3$ candies.

In the whole numbers, both these are represented as $3$$3$.

In integers, the first is $+3$$+ 3$ and the second is $-3$$- 3$.

Integer numbers are represented as follows.

$3$$3$ is represented as either $\text{received:}3$$\textrm{\left(r e c e i v e d\right\rangle} 3$ or \text{aligned:}3$\textrm{\left(a l i g \ne d\right\rangle} 3$.

$-3$$- 3$ is represented as either $\text{given:}3$$\textrm{\left(g i v e n\right\rangle} 3$ or $\text{opposed:}3$$\textrm{\left(o p p o s e d\right\rangle} 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$$8$ pieces.

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

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

The same in fractions is $3\frac{5}{8}$$3 \frac{5}{8}$.

fractions are non-standard

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

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

To multiply the two fractions $\frac{2}{6}$$\frac{2}{6}$ and $\frac{1}{14}$$\frac{1}{14}$, multiply the numerators and multiply the denominators $\frac{2×1}{6×14}$$\frac{2 \times 1}{6 \times 14}$, which results in completely different form $\frac{2}{84}$$\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$$1 / 10$

•  one hundredth or $1/100$$1 / 100$

•  one thousandth or $1/1000$$1 / 1000$

•  etc.

By this, a fraction $\frac{1}{2}$$\frac{1}{2}$ is given as $\frac{5}{10}$$\frac{5}{10}$.

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

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

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

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

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

The place value of $5$$5$ in $0.75$$0.75$ is "hundredth"

Note that the number is given equivalently as $\frac{75}{100}$$\frac{75}{100}$ which equals $\frac{7}{10}+\frac{5}{100}$$\frac{7}{10} + \frac{5}{100}$. The decimal representation $0.75$$0.75$ is understood as $\frac{7}{10}+\frac{5}{100}$$\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$$10$.

A mixed fraction $r\frac{p}{q}=r+\frac{a}{10}+\frac{b}{100}+\frac{c}{1000}+\cdots$$\textcolor{c \mathmr{and} a l}{r \frac{p}{q} = r + \frac{a}{10} + \frac{b}{100} + \frac{c}{1000} + \cdots}$ is given as $r.abc\cdots$$\textcolor{c \mathmr{and} a l}{r . a b c \cdots}$ in decimal representation.

Example:

$32\frac{5}{8}=30+2+\frac{6}{10}+\frac{2}{100}+\frac{5}{1000}$$32 \frac{5}{8} = 30 + 2 + \frac{6}{10} + \frac{2}{100} + \frac{5}{1000}$

$=32.625$$= 32.625$

examples

What is the decimal representation of $\frac{3}{4}$$\frac{3}{4}$?
The answer is "$0.75$$0.75$".

$\frac{3}{4}$$\frac{3}{4}$
$=\frac{75}{100}$$= \frac{75}{100}$
$=\frac{7}{10}+\frac{5}{100}$$= \frac{7}{10} + \frac{5}{100}$
$=0.75$$= 0.75$

What is the decimal representation of $\frac{2}{25}$$\frac{2}{25}$?
The answer is "$0.08$$0.08$".

$\frac{2}{25}$$\frac{2}{25}$
$=\frac{8}{100}$$= \frac{8}{100}$
$=\frac{0}{10}+\frac{8}{100}$$= \frac{0}{10} + \frac{8}{100}$
$=0.08$$= 0.08$

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

Note that $\frac{1}{3}=\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}+\cdots$$\frac{1}{3} = \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \cdots$.
The answer is "$0.333\cdots$$0.333 \cdots$".

summary

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

The place-value of decimals is standardized to be

•  one tenth or $1/10$$1 / 10$

•  one hundredth or $1/100$$1 / 100$

•  one thousandth or $1/1000$$1 / 1000$

•  etc. eg: $\frac{3}{4}$$\frac{3}{4}$
$=\frac{75}{100}$$= \frac{75}{100}$
$=\frac{7}{10}+\frac{5}{100}$$= \frac{7}{10} + \frac{5}{100}$
$=0.75$$= 0.75$ $7$$7$ has tenth ($\frac{1}{10}$$\frac{1}{10}$) place value and $5$$5$ has hundredth ($\frac{1}{100}$$\frac{1}{100}$) place value.

Outline