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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.
mathsWhole NumbersComparing Whole Numbers

### Comparing two Whole Numbers

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

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

A simplified procedure based on place-value of large numbers is explained.

click on the content to continue..

Whole numbers are used to represent count or measure of quantities.

Comparing the number of fish and cats in the figure, it is found that, number of fish is more than number of cats.

That is 10 is larger than 5. .

number of cars is equal to the number of cats.

That is 5 equals 5.

number of cats is less than number of fish.

That is 5 is less than 10

Whole numbers are used to measure quantities. An example is height of a tree.

The pine tree, shown in picture, is taller than the palm tree shown in the picture.

Such measurement of quantities or numbers can also be compared for smaller and equal.

One of the following is true while comparing two quantities.

•  A quantity is lesser than another.

•  A quantity equals another.

•  A quantity is more than another.

For example,
•  4 is less than 5, which is given as 4 < 5

•  5 equals 5, which is given as 5=5

•  5 is more than 4, which is given as 5>4 When compared to a number, another number can only be one of the three,
(1) smaller,
(2) equal, or
(3) larger

3<7

The symbol < is used to denote the number on the left side is less than the number on the right. In this case, 3 is less than 7.

7>3

The symbol > is used to denote the number on the left side is greater than the number on the right. In this case, 7 is greater than 3.

3=3

The symbol = is used to denote the number on the left side equals the number on the right. In this case, 3 equals 3.

Considering numbers 7 and 8. There are 7 burgers and 8 dogs in the figure.

Consider comparing 8 and 7. The comparison of numbers as quantities is illustrated in the figure. In first principles, two quantities are matched one-to-one. The quantity that has excess is greater. The other quantity is smaller. If the two quantities match to the last count, then they are equal. The comparison of numbers as quantities is illustrated in the figure. In first principles, two quantities are matched one-to-one.

Number of dogs have one excess, so number of dogs is greater than the number of burgers.

Consider comparing 8 and 7. The comparison of numbers as quantities is illustrated in the figure.

The quantity that has excess is greater. And

The the other quantity is lesser.

Comparison of two numbers 19 and 17 is illustrated in the figure.

On comparing the quantities, 17 is found to be smaller.

Comparing 19 and 17. By first principles: The quantities are matched. The tens is made of 10 units. So first the tens, given in purple bar, are matched one-to-one. Then the units, given in blue cubes, are matched and found that 19 has excess. So 17 is found to be smaller.

As a Simplified Procedure : Comparison by place-value , the numbers in tens place value can be compared. 1 in 19 and 1 in 17. Both digits in tens place are equal. Then the numbers in the units position are compared. 9 in the 19 and 7 in 17. On comparing, 17 is found to be smaller.

Solved Exercise Problem:

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. A ten is made of 10 units. The tens, given in purple bar, are matched one-to-one. From this, 53 is found to be larger.

As a simplified procedure : Comparison by place-value, the numbers in tens place value can be compared. 4 in 47 and 5 in 53. On comparing, 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.

Solved Exercise Problem:

Compare numbers 214 and 132. Which one is larger?

• 214
• 214
• 132

The answer is "214"

Comparing numbers 214 and 132. By First principles, the two quantities are matched. The hundreds are the largest among the three and made of 10 tens. So the hundreds are matched. From this, 214 is found to be larger.

As a simplified procedure: Comparison by place-value, the numbers in hundreds place are compared first. 2 from 214 is compared with 1 from 132. Since 2 is greater than 1, it is concluded that the 214 is larger.

Comparing numbers 214 and 132. The numbers are given in the place-value form. Comparing the hundreds place, 2 is greater than 1, it is concluded that the 214 is larger.

Solved Exercise Problem:

Compare numbers 156 and 88. Which one is smaller?

• Comparing 1 of 156 and 8 of 88, 156 is smaller
• Comparing 5 of 156 and 8 of 88, 156 is smaller
• Comparing 1 of 156 and 0 of 088, 88 is smaller
• Comparing 1 of 156 and 0 of 088, 88 is smaller

The answer is "Comparing 1 of 156 and 0 of 088, 88 is smaller"

Comparing 156 and 88. It is noted that the hundreds place is 0 for 88.

By the simplified procedure, the digits in the hundreds place are compared and 0 is smaller than 1. It is concluded that 88 is smaller.

familiarize with the terminology
greater than

familiarize with the terminology
less than

familiarize with the terminology
equals

Solved Exercise Problem:

Which of the following is larger than the other? 23 or 61

• 23
• 61
• 61

The answer is "61". 61 > 23.

Solved Exercise Problem:

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

• 2183
• 2183
• 2661
• both of them

The answer is "2183". 2183 < 2661.

Solved Exercise Problem:

Which of the following is greater than the other? 341 or 341

• first 341
• second 341
• they are equal
• they are equal

The answer is "the numbers are equal". 341 = 341

Comparison of whole numbers : Two numbers can be compared to find one of them as

•  smaller or less

•  equal

•  greater or larger When compared to a number, another number can only be one of the three,

(1) smaller,

(2) equal, or

(3) larger

Comparison by First Principle: To find if one number is larger or smaller than another number, the quantities represented by them are compared. Example: Comparing the numbers 47 and 53. The quantities represented by them is compared in the figure. It is found that 47 is smaller than 53.

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. For example: Let us take the numbers 2471 and 2437.

Largest place value is 1000 and the digits are 2 and 2, which are equal.

Next smaller place value is 100 and the digits are 4 and 4, which are equal.

Next smaller place value is 10 and the digits are 7 and 3, in which, 7 is greater than 3.

So 2471 is greater than 2437.

Solved Exercise Problem:

Which of the following step help to find the larger between 23 and 156?

• compare first digits 2 and 1
• compare the 100s place 0 and 1
• compare the 100s place 0 and 1

The answer is "compare the 100s place 0 and 1 ". The numbers are equivalently given as 023 and 156.

Solved Exercise Problem:

Find the smaller: 2323 and 99?

• 2323
• 99
• 99

The answer is "99". Comparing the 1000s place value 2 and 0, the smaller is found.

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