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

• multiplication in first principles -- repeatedly combining a quantity and measuring the combined and

• simplified procedure : Sign property of multiplication and Multiplication by Place value for Decimals.

*click on the content to continue..*

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

- multiplicand repeated multiplier number of times
- multiplicand repeated multiplier number of times
- multiplier is taken-away from multiplicand

The answer is "multiplicand repeated multiplier number of times".

What are "decimals"?

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

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

What is `0.2 xx 0.3`?

- `0.6`
- `0.06`
- `0.06`

The answer is "`0.06`".

By first principles, the multiplicand `0.2 = 2/10` is repeated multiplier `0.3=3/10` number of times.

This is done in two steps,

first step: `2/10` repeated `1/10` (denominator of `3/10`) times is `2/100`.

Second step: the result from the first step `2/100` is repeated `3` (numerator of `3/10`) times is `2/100 + 2/100 + 2/100` `=6/100`

The product is `6/100`, or equals `0.06` 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 `0.2 xx (-0.3)`?

- `0.06`
- `-0.06`
- `-0.06`

The answer is "`-0.06`".

By first principles, the multiplier `0.2 = 2/10`*aligned in direction* is repeated multiplier `0.3=3/10`*opposed in direction* number of times.

This is done in two steps,

first step: `2/10` repeated `1/10` (denominator of `3/10`) times is `2/100`.

Second step: the result from the first step `2/100` is repeated `3` (numerator of `3/10`) times*opposed in direction* is `2/100 + 2/100 + 2/100` `=6/100`*opposed in direction*.

The product is `6/100`*opposed in direction*, or equals `-0.06` in decimals.

**Decimal multiplication by first principles** : Decimal multiplication is repeating the multiplicand, multiplier number of times with sign of the numbers (direction) handled appropriately.

In whole numbers, we have studied *Multiplication 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 Multiplication*.

+ve `xx` +ve = +ve

+ve `xx` -ve = -ve

-ve `xx` +ve = -ve

-ve `xx` -ve = +ve

This is applicable to decimals.

In Fractions, we have studied *Multiplication of Numerators and Denominators*.

For example, to multiply `4/5 xx 3/2`, the numerators are multiplied and denominators are multiplied. The product is `(4 xx 3)/(5 xx 2)`.

Similarly decimals are multiplied keeping in mind the place-value representation.

For example, to multiply `0.8 xx 1.5`, it is equivalently thought as `8/10 xx 15/10` and so the product is `(8xx15)/(10xx10)`. Note that `8xx15` is multiplied as per "multiplication by place-value".

Multiply `0.007 xx 0.05`?

- `0.035`
- `0.0035`
- `0.00035`
- `0.00035`
- `0.000035`

The answer is "0.00035". This is explained in the next page.

Consider multiplication of `0.007 xx 0.05`

This is equivalently `7/1000 xx 5/100`

`=(7 xx 5)/(1000xx 100)`

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

The decimal point of multiplicand and multiplier are removed and the numbers are multiplied as integers.

eg: `0.007` is modified to the integer form `7`.

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

The number of decimal-places in the multiplicand and multiplier are counted.

eg: `0.007` has `3` decimal-places.

`0.05` has `2` decimal-places.

Now the integer forms are multiplied.

eg: `7xx5 = 35`

The number of decimal places of multiplicand and multiplier are added.

eg: `3+2=5`.

The product of integer forms is modified to have the number of decimal points give by the sum above.

eg: `35` is modified to `0.00035` Which has `5` decimal points.

What is `12.55 xx 0.002`

- `25.1`
- `0.02510`
- `0.02510`

The answer is "`0.02510`".

`1255xx2 = 2510.`

Total number of decimal places in the multiplicand and multiplier is `2+3=5`

So the product decimal place moves `5` places

`12.55 xx 0.002 = 0.02510`

**Decimal Multiplication -- Simplified Procedure** : The signs (+ve / -ve) are handled as in *Sign-property of Integer Multiplication*

• +ve `xx` +ve = +ve

• +ve `xx` -ve = -ve

• -ve `xx` +ve = -ve

• -ve `xx` -ve = +ve

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

• The decimal place is re-introduced in to the product.

• The decimal-point is moved to the left -- a number of digits equal to the total number of decimals in multiplier and multiplicand

*slide-show version coming soon*