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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.

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mathsIntegersComparing Integers

Comparing Integers

In this page, comparing two integers to find one as larger, equal or smaller is explained.

The comparison in first principles is explained for directed numbers and it is extended to comparison as ordered sequence.

A simplified procedure by sign and place-value of large numbers is explained.



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In whole numbers, we had studied the following.

Trichotomy Property of Comparison : Two numbers can be compared to find one of them as trichotomy property

 •  smaller or less

 •  equal

 •  greater or larger

Comparison by First Principle: Two quantities are matched one-to-one and compared in the count or magnitude of the quantities. As a result one of them is smaller or equal to or larger to the other. whole number comparison Example: Comparing the numbers `7` and `8`. The quantities represented by them is compared in the figure. It is found that `7` is smaller than `8`.

Simplified Procedure -- Comparison by ordered-sequence: To find if one numbers is larger or smaller than another number, the numbers are compared using the order `0,1,2,3,4,5,6,7,8,9, 10, cdots`. comparing on number line Example: Comparing the numbers `4` and `7`.

`4` is on the left-side to `7` in the order `0,1,2,3,4,5,6,7,8,9` and so `4` is smaller than `7`.

Comparison by Place-value -- Simplified Procedure to Compare Large Numbers: To find if one number is larger or smaller than another number, the digits at the highest place value are compared and if they are equal, then the digits at next lower place value are compared. comparing numbers simplified procedure

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

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

The answer is "`6`".

The given two numbers are whole numbers. The same is understood in integers as `text(received:)3` and `text(received:)6`.

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

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

The answer is "`3`".

`text(received:)3` is larger than `text(given:)6`.

To understand how `text(received:)3` is larger than `text(given:)6`, consider the box-of-candy example.

A girl and her brother each have a box of candies. The number of candies in the box is not counted and they have equal numbers to start with.

She received `3` candies.
Her brother gave `6` candies.

Who would have more candies?
Obviously the sister as the number of candies in her box increased. The number of candies in brother's box decreased because he gave away some of the candies he had.

Consider the numbers `text(received:)3` and `text(received:)6`. `text(received:)6` is larger than `text(received:)3`. The comparison is in terms of amount received.

Consider the numbers `text(received:)3` and `text(given:)6`. In this, `text(given:)6` gives a value opposite of amount received. When comparing in terms of amount received, `text(received:)3` is larger than `text(given:)6`.

Which of the following is a generalized statement of the above?

  • negative numbers are smaller than positive numbers
  • negative numbers are smaller than positive numbers
  • no generalization possible

The answer is "negative numbers are smaller than positive numbers".

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

comparison is in terms of received. and
`-3 = text(given:)3`
`-6 = text(given:)6`

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

The answer is "`-3`".

In terms of amount received, `text(given:)3` is larger than `text(given:)6`.

To understand how `text(given:)3` is larger than `text(given:)6`, consider the box-of-candy example.

A girl and her brother each have a box of candies. The number of candies in the box is not counted and they have equal numbers to start with.

She gave `3` candies.
Her brother gave `6` candies.

Who has more candies?
Sister has more candies. The number of candies in the sister's box is reduced a smaller amount. The number of candies in brother's box is reduced a larger amount.

Consider the numbers, `-3 = text(given:)3` and `-6 = text(given:)6` .

When comparing in terms of amount received, `text(given:)3` is larger than `text(given:)6`.

Which of the following is a generalized statement of the above?

  • negative number with large amount is smaller than negative number with smaller amount
  • negative number with large amount is smaller than negative number with smaller amount
  • no generalization possible

The answer is "negative number with large amount is smaller than negative number with smaller amount". Amount is the absolute value of the integer.

Comparison of Integers -- First Principles: : Comparison is in terms of the amount received.

Amount given is smaller than amount received, as comparison is by amount received.

Larger amount given is smaller than smaller amount given, as comparison is by the amount received.

Comparison of Integers -- Simplified Procedure:

sign-property of comparison
 •  +ve and +ve are compared as whole numbers.
 •  When comparing +ve and -ve, the +ve value 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.

The absolute values are compared as the simplified procedure detailed in whole numbers comparison by place-value.

Solved Exercise Problem:

Which of the following is smaller than the other? `21` or `-26`

  • `21`
  • `-26`
  • `-26`
  • both of them

The answer is "`-26`." As per sign property of comparison, the negative number is smaller than the positive number.

Solved Exercise Problem:

Which of the following is smaller than the other? `-2183` or `-2661`

  • `-2183`
  • `-2661`
  • `-2661`
  • both of them
  • neither of them

The answer is "`-2661`". Both the numbers are negative. So, the larger amount in negative is smaller in values.

Comparing the numbers without the sign and using the comparison by place-value, 2 thousands equals 2 thousands. 6 hundreds is larger than 1 hundreds. So, 2661 is larger, and negative 2661 is smaller.

Solved Exercise Problem:

Which of the following is larger than the other? `-341` or `-341`

  • first `-341`
  • second `-341`
  • they are equal
  • they are equal

The answer is "they are equal"

Solved Exercise Problem:

Which of the following step help to find the larger among `-23` and `-156`?

  • compare the signs to decide
  • compare the absolute values `23` and `156`
  • compare the absolute values `23` and `156`

The answer is "compare the absolute values `23` and `156`". The smaller absolute value is the larger number.

Solved Exercise Problem:

Find the smaller number among `-2323` and `-99`?

  • `-2323`
  • `-2323`
  • `-99`

The answer is "`-2323`".

                            
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