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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 Division

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.



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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.decimal division 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.

                            
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