Decimals : Standard form of Fractions: Decimals

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Standard form of Fractions: Decimals

what you'll learn...

overview

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

$0,1,2,cdots$ 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 $received: 3$ $text(received:)3$ or $aligned: 3$ $text(aligned:)3$ .

$- 3$ $-3$ is represented as either $given: 3$ $text(given:)3$ or $opposed: 3$ $text(opposed:)3$ .

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

fractions are non-standard

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

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

To multiply the two fractions $\frac{2}{6}$ $2/6$ and $\frac{1}{14}$ $1/14$ , multiply the numerators and multiply the denominators $\frac{2 \times 1}{6 \times 14}$ $(2xx1)/(6xx14)$ , which results in completely different form $\frac{2}{84}$ $2/84$ .

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

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

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

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 $r \frac{p}{q} = r + \frac{a}{10} + \frac{b}{100} + \frac{c}{1000} + \dots$ $color(coral)(r p/q = r + a/10 + b/100 + c/1000 + cdots)$ is given as $r . a b c \dots$ $color(coral)(r.abc cdots)$ in decimal representation.

Example:

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

$= 32.625$ $= 32.625$

examples

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

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

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

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

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

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

summary

Fractions are represented in various non-standard representations. $\frac{p}{q}$ $p/q$ and $\frac{l}{m}$ $l/m$ are two fractions given in two different place value $\frac{1}{q}$ $1/q$ and $\frac{1}{m}$ $1/m$ 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: $\frac{3}{4}$ $3/4$
$= \frac{75}{100}$ $=75/100$
$= \frac{7}{10} + \frac{5}{100}$ $=7/10 + 5/100$
$= 0.75$ $=0.75$ $7$ $7$ has tenth ( $\frac{1}{10}$ $1/10$ ) place value and $5$ $5$ has hundredth ( $\frac{1}{100}$ $1/100$ ) 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