Decimals are fractions with standard place values. In this page, the following for decimals are explained.

• Division in first principles -- splitting a quantity into a number of parts and measuring one part

• simplified procedure : Sign property of division and long division method for Decimals.

*click on the content to continue..*

Which of the following is a description of "division"?

- dividend split into divisor number of parts
- division is inverse of multiplication
- both the above
- both the above

The answer is "both the above".

What is a "decimal"?

- decimals are fractions with standardized place-values
- decimals are fractions with standardized place-values
- a group of `10` animals

The answer is "decimals are fractions with standardized place-values".

What is a `0.6 -: 3`?

- `0.2`
- `0.2`
- `2`

The answer is "`0.2`".

By first principles, the dividend `0.6 = 6/10` is split into divisor `3` number of equal parts : `2/10 ; 2/10 ; 2/10`. And one part is `2/10`, or equals `0.2` in decimals.

What is a `0.6 -: 0.3`?

- `0.2`
- `2`
- `2`

The answer is "`2`".

By first principles, the dividend `0.6 = 6/10` is split into divisor `0.3 = 3/10` number of equal parts :

This is done in two steps:

first step: `6/10` is split into `1/10` (denominator of `3/10`) times is `60/10`.

Second step: the result from the first step `60/10` is split into `3` (numerator of `3/10`) times is `20/10`

The result is `20/10`, or equals `2` in decimals.

Are decimals directed numbers?

- No, decimals are only positive
- Yes, decimals can be positive or negative
- Yes, decimals can be positive or negative

The answer is "Yes, decimals can be positive or negative".

What is `2`*aligned in direction* in integer form?*quickly revising directed numbers aligned and opposing from integers*

- `+2`
- `+2`
- `-2`

The answer is "`+2`".

What is `2`*opposed in direction* in integer form?*quickly revising directed numbers aligned and opposing from integers*

- `+2`
- `-2`
- `-2`

The answer is "`-2`". Directed numbers, positive and negative, are explained as "aligned in direction" and "opposed in direction", respectively.

What is `0.6 -: (-0.3)`?

- `+2`
- `-2`
- `-2`

The answer is "`-2`".

By first principles, the dividend `0.6 = 6/10`*aligned in direction* is split into the divisor `0.3=3/10`*opposed in direction* number of equal parts :

This is done in two steps,

first step: `6/10`*aligned in direction* is split into `1/10`*aligned in direction* (denominator of `3/10`) times is `60/10`*aligned in direction*.

Second step: the result from the first step `60/10`*aligned in direction* is split into `3` (numerator of `3/10`)*opposed in direction* times is `20/10`*opposed in direction*.

The result is `20/10`*opposed in direction*, or equals `-2` in decimal number form.

**Decimal Division by first principle** : Decimal division is splitting the dividend, into divisor number of parts with sign of the numbers (direction) handled appropriately.

In whole numbers, we have studied *Division by Place-value* as illustrated in the figure. This procedure is used in decimals in a later step. In Integers, we have studied *Sign-property of Division*.

• +ve `-:` +ve = +ve

• +ve `-:` -ve = -ve

• -ve `-:` +ve = -ve

• -ve `-:` -ve = +ve

This is applicable to decimals.

In Fractions, we have studied that *division is inverse of multiplication*.

For example, to divide `4/5 -: 3/2`, it is modified to `4/5 xx 2/3`

Decimals are divided keeping in mind the place-value representation, which is a form of fractions.

For example, to divide `1.8 -: .06`, it is equivalently thought as `18/10 -: 6/100` and modified to multiplication `18/10 xx 100/6`.

Divide `0.002 -: 0.05`?

- `0.04`
- `0.04`
- `0.004`
- `0.0004`

The answer is "`0.04`". This is explained in the next page.

Consider division of `0.002 -: 0.05`

This is equivalently `2/1000 -: 5/100`

`=2/1000 xx 100/5`

`=(2xx100)/(1000xx 5)`

`=2/50`

`=0.04`

Understanding the above, a simplified procedure to divide the decimals is devised.

The decimal point of dividend and divisor are removed and the numbers are divided as integers.

eg: `0.002` is modified to the integer form `2`.

`0.05` is modified to the integer form `5`.

The number of decimal-places in the dividend and divisor are counted.

eg: `0.002` has `3` decimal-places.

`0.05` has `2` decimal-places.

Now the integer forms are divided.

eg: `2-:5 = 0.4`

The number of decimal places of divisor is subtracted from dividend number of decimal places.

eg: `3-2=1`.

The quotient form is modified to have the number of decimal points give by the difference above.

eg: `0.4` is modified to `0.04`, that is decimal point moved to the left by `1`.

What is `1.55 -: .005`

- `3.1`
- `310`
- `310`

The answer is "`310`".

`155-: 5 = 31.`

Difference in decimal places is `2-3=-1`

So the product decimal place moves `1` place to the right.

`1.55 -: 0.005 = 310`

**Decimal Division -- Simplified Procedure** :

The signs (+ve / -ve) are handled as in *Sign-property of Integer Division*

• +ve `-:` +ve = +ve

• +ve `-:` -ve = -ve

• -ve `-:` +ve = -ve

• -ve `-:` -ve = +ve

The decimal places are removed and the division is carried out as per *Whole number Division by Place Value*.

• The decimal-point is modified in the result

• The decimal-point is moved to the left the number of times equal to the difference number of decimals in dividend minus number of decimals in divisor. A positive difference moves the decimal point to the left, and a negative difference moves the decimal point to the right.

*slide-show version coming soon*