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Thought-Process to Discover Knowledge

Welcome to nubtrek.

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

Comparing Decimals

Comparing two decimals is explained.

Comparison in first principles -- matching two quantities to find one as smaller, equal, or larger than the other -- is extended for decimals (fractions of standard place values).

Based on this a simplified procedure by place value is explained.



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Given numbers `7` and `3`, which number is larger?

  • `7`
  • `7`
  • `3`

The answer is '`7`'

Given numbers `4` and `6`, which number is smaller?

  • `6`
  • `4`
  • `4`

The answer is '`4`'

Given numbers `5` and `5`, which number is smaller?

  • One of them has to be smaller
  • The numbers are equal
  • The numbers are equal

The answer is 'The numbers are equal'

Compare numbers `47` and `53`. Which one is larger?

  • `47`
  • `53`
  • `53`

The answer is "`53`".

Comparing the numbers `47` and `53`.

By first principles : The quantities are matched. On comparing the quantities, `53` is found to be larger.

As a simplified procedure, the numbers in tens place value can be compared. `4` in `47` and `5` in `53`. On comparing the tens place, `53` is found to be larger.

Note that once it is found that the tens place are not equal, the larger number is decided. The units place does not need to be compared. Is it correct?

  • Yes, Because the tens place has `10` pieces of one piece in units place.
  • Yes, Because the tens place has `10` pieces of one piece in units place.
  • No, the units place should also be compared.

The answer is "Yes, Because the tens place has `10` of units place".

This simplified procedure, "Comparison by Place-value" was introduced in whole numbers.

Which of the following is larger? `3` or `-6`

  • `3`
  • `3`
  • `-6`
  • both of them

The answer is "`3`". `text(aligned:)3` is larger than `text(opposed:)6`.

Given two fractions `3/4` and `5/8`. Which fraction is larger?

  • comparing numerators `3` and `5`, it is clear that `5/8` is larger
  • considering the place values are different, the fractions cannot be compared without converting them to like fractions
  • considering the place values are different, the fractions cannot be compared without converting them to like fractions

The answer is 'considering the place values are different, the fractions cannot be compared without converting them to like fractions'.

Given two fractions `3/4` and `5/8`. The fraction `3/4` is converted to `6/8` to make them like fractions.

Both `6/8` and `5/8` have the same place value `1/8` (ie: same denominator `8`).

In this form, the numerators can be compared and `6/8` is larger, which means its equivalent fraction `3/4` is larger than `5/8`.

Given two whole numbers, integers, or fractions, they can be compared as

 •  greater or larger

 •  smaller or lesser

 •  equal.

Comparison by First-principles : Two numbers representing count or quantities are matched. The amount that is more than other is larger, and the other number is smaller. If the amounts match then the numbers are equal.

Comparison by Place-value : For large number with units, tens, hundreds, ... digits, the comparison is started from the largest place value till one of the number is found to be larger or smaller.

Sign Property of comparison : To compare integers with positive and negative signs, the direction is also taken into consideration.

Comparison of Fractions :To compare fractions, they are converted to like fractions and the numerators are compared like an integer.

What are decimals?

  • decimals are standardized fractions with place value of fractions as power of `10`
  • decimals are standardized fractions with place value of fractions as power of `10`
  • decimals are `10, 100, 1000, cdots`

The answer is "decimals are standardized fractions with place value of fractions as power of `10`".

Given two decimals `0.7` and `0.6`, which one is larger?

  • `0.7`
  • `0.7`
  • `0.6`

The answer is "`0.7`".

It is noted that
  `0.7` means `7` counts of `1//10^(th)` parts and
  `0.6` means `6` counts of `1//10^(th)` parts

So `7` counts or `0.7` is larger.

Given two decimals `0.7` and `0.62`, which one is larger?

  • `0.7`
  • `0.7`
  • `0.62`

The answer is "`0.7`".

It is noted that
  `0.7` means `7` counts of `1//10^(th)` parts and
  `0.62` means `6` counts of `1//10^(th)` parts and `2` counts of `1//100^(th)` parts.

So `7` counts or `0.7` is larger.

Given two decimals `0.7` and `1.2`, which decimal is larger?

  • `0.7`
  • `0.7`
  • `1.2`

The answer is "`1.2`".

It is noted that
  `0.7` is `7` counts of `1//10^(th)` parts and
  `1.2` is `1` whole and `2` counts of `1//10(th)` parts.

So `1` count of whole or `1.2` is larger.

Given two decimals `-0.7` and `0.62`, which decimal is larger?

  • `-0.7`
  • `0.62`
  • `0.62`

The answer is "`0.62`".

It is noted that
  `-0.7` represents `text(opposed:)7` counts of `1//10^(th)` parts and
  `0.62` represents `text(aligned:)6` counts of `1//10^(th)` parts and `text(aligned:)2` counts of `1//100(th)` parts.

So, the positive value `0.62` is larger.

Given two decimals `-0.5` and `-0.2`, which decimal is larger?

  • `-0.5`
  • `-0.2`
  • `-0.2`

The answer is "`-0.2`".

It is noted that
  `-0.5` means `text(opposed:)5` counts of `1//10^(th)` parts and
  `-0.2` means `text(opposed:)2` counts of `1//10^(th)` parts.

So, negative of smaller absolute value `-0.2` is larger.

In whole numbers, we learned that the place-value order is
units `<` tens `<` hundreds `< cdots`.

In decimals, the place-value is extended with `1/10`, `1/100`, etc.

What is the revised place-value order for decimals?

  • units `< 1//10 <` tens `< 1//100 <` hundreds, etc.
  • `cdots <1//100 < 1//10 < ` units `<` tens `<` hundreds `< cdots`
  • `cdots <1//100 < 1//10 < ` units `<` tens `<` hundreds `< cdots`

The answer is "`cdots <1//100 < 1//10 < ` units `<` tens `<` hundreds `< cdots`".

The place-value order is extended for decimals as in
`cdots <1//100 < 1//10 < ` units `<` tens `<` hundreds `< cdots`

Which of the following is larger than the other? `-3.41` or `-3.41`

  • first `-3.41`
  • second `-3.41`
  • they are equal
  • they are equal

The answer is "they are equal"

Comparison of Decimals :
Decimals are standardized fractions of whole numbers. Comparison based on place-value is applicable. The place value is extended to `1//10^(th)`, `1//100^(th)`, etc.

Decimals are directed numbers. Sign-property of comparison (as learned for integers) is applicable.
 •  When comparing +ve and +ve, the number with larger value is larger than the other.
 •  When comparing +ve and -ve, the +ve number is larger irrespective of the absolute values of the numbers.
 •  When comparing -ve and -ve, the number with smaller absolute value is larger than the other.

Comparison by place-value (as learned for whole numbers) is applicable with extended place-value order.
`cdots <1//100 < 1//10 < ` units `<` tens `<` hundreds `< cdots`

Which of the following step help to find the larger among `-.23` and `-1.56`?

  • compare the first digit `-2` and `-1`
  • compare the absolute values `.23` and `1.56`
  • compare the absolute values `.23` and `1.56`

The answer is "compare the absolute values `.23` and `1.56`". The smaller absolute value is the larger number.

Which of the following is smaller than the other? `-21.83` or `-2.661`

  • `-21.83`
  • `-21.83`
  • `-2.661`

The answer is "`-21.83`".

Find the smaller number among `-2.3` and `-.0099`

  • `-2.3`
  • `-2.3`
  • `-.0099`

The answer is "`-2.3`".

We have learned about comparing decimals. This can be used to order more than two decimals in ascending or descending orders. Let us quickly review these.

What does the word 'ascend' mean? ascending meaning

  • to go up or climb
  • to go up or climb
  • to write a reason for

The answer is 'to go up or climb'.

What is the order called when fractions are ordered from the smallest to the largest?ascending meaning

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is 'ascending order'.

What does the word 'descending' mean? descending meaning

  • to move down
  • to move down
  • to write a reason for

The answer is 'to move down'.

What is the order called when fractions are ordered from the largest to the smallest?descending meaning

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is 'descending order'.

Arrange the numbers in ascending order `-23.4`, `7.3`, `42.3`.

  • `7.3`, `-23.4`,`42.3`
  • `-23.4`, `7.3`, `42.3`
  • `-23.4`, `7.3`, `42.3`

The answer is "`-23.4`, `7.3`, `42.3`"

Arrange the numbers in descending order `8.73`, `.9`, `-9.9`.

  • `8.73`, `.9`, `-9.9`
  • `8.73`, `.9`, `-9.9`
  • `-9.9`, `8.73`, `.9`

The answer is "`8.73`, `.9`, `-9.9`"

Two or more decimals can be compared to arrange them in
 •  ascending order : from the smallest to the largest
 •  descending order : from the largest to the smallest

Arrange the numbers in ascending order `-8`, `-8.43`, `-8.4`.

  • `-8.43`, `-8.4`, `-8`
  • `-8.43`, `-8.4`, `-8`
  • cannot order, as one of the numbers is not a decimal number

The answer is "`-8.43`, `-8.4`, `-8`".

Arrange the numbers in descending order `0.002`, `0.02`, `0.0002`.

  • `0.0002`, `0.002`, `0.02`
  • `0.02`, `0.002`, `0.0002`
  • `0.02`, `0.002`, `0.0002`

The answer is "`0.02`, `0.002`, `0.0002`"

Arrange the numbers in descending order `0.5`, `-5`, `5`, `-0.5`.

  • `5`, `-5`
  • `5`, `.5`, `-0.5`, `-5`
  • `5`, `.5`, `-0.5`, `-5`

The answer is "`5`, `.5`, `-.5`, `-5`"

Arrange the numbers in ascending order `-2.3`, `0`, `4.2`.

  • `-2.3, 0, 4.2`
  • `-2.3, 0, 4.2`
  • `0, -2.3, 4.2`

The answer is "`-2.3, 0, 4.2`"

                            
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