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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsDecimalsDecimal Arithmetics

Decimal Multiplication

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.



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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`) timesopposed 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.multiplication by place value 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