Comparing Large Numbers: Definition, Diagram, Rules and Examples

Comparing Large Numbers: In our daily life, we often need to compare numbers. Some numbers are large, some are very large, and some are small and very small. In mathematics, different methods are used to compare numbers of different magnitudes. In this article, we shall focus our discussion on comparing large numbers only.

There are some basic symbols used for the comparison of numbers: they are greater than \((>)\), less than \((<)\), or equal sign \((=)\).

Define Large Numbers

Definition: In a number system, numbers that are ordinarily bigger or greater than the other numbers are large.
Example: \(1\) lakh, \(1\) million etc., come under the category of large numbers.

What is a Number System?

The number system is the writing system of expressing the numbers, a mathematical notation for representing the numbers of a given set, by consistently using the numbers or symbols.

Imaginary Numbers

A number that does not exist on the number line is imaginary. For example, The square roots of negative numbers are imaginary numbers. It is denoted by ‘\(i\)’ or ‘\(j\)’.

Real Numbers

The combination of rational and irrational numbers is known as real numbers. Real numbers can be both positive or negative, which are denoted as ‘\(R\)’. Natural numbers, fractions, decimals all come under this category.

Integers

All numbers which do not have the decimal place in them are known as integers. 

\(Z = \left\{ { – \infty ……. – 3, – 2, – 1,0,,1,2,3…..\infty } \right\}\)

Positive integers: \(1, 2, 3,…..\) 4 are the set of all positive integers. 

Negative Integers: \(-1, -2, -3….\) are the set of all negative integers. 

Non-Positive and Non-Negative Integers: \(0\) is neither the positive integer nor the negative integer.

Natural Numbers

When we count objects in a group of things, we start counting from one and then go on to two, three, four etc. this is a natural way of counting objects. The counting numbers \(1, 2, 3, 4, 5,…\) are known as natural numbers. 

Whole Numbers

The numbers \(1, 2, 3, 4, …\) etc., are natural numbers. These natural numbers, along with the number zero, form the collection of whole numbers. That is, numbers \(0, 1, 2, 3, …\) are called whole numbers.

Large Numbers

You are well aware that numbers are put into various groups like tens, hundreds, thousands, millions, billions, etc. Each group is home to three subdivisions: ones, tens, and hundreds. When reading or writing, a large number begins at the left with the largest group and proceed to the right.

For instance, take \(9,654\) is read as nine thousand, six hundred, and fifty-four. The first one must be clear with this. It begins with the ones: ones, tens and hundreds being the subdivisions, followed by thousands, millions, and billions and so on, with their own three subdivisions. 

How would you read \(13,965,291?\) It would go thirteen million, nine-hundred sixty-five thousand, and two hundred and ninety-one.

The very large numbers are somewhat like this:

\(1,000,000,000\) are “one billion”.
\(1,000,000,000,000\) are “one thousand billion”.
\(1,000,000,000,000,000\) are “one million billion”.
\(1,000,000,000,000,000,000\) are “one billion billion”.
\(1,000,000,000,000,000,000,000\) are “a thousand billion billion” and so on…

What is Comparison of Numbers?

Comparison tells us the similar properties of various objects. It is the primary concept in mathematics that helps us describe whether the numbers are the same, or one is greater than or one is smaller than the other, in comparing two numbers.

We have mainly three special symbols that are used for comparing the numbers. The basic symbols used in the comparison of numbers are given below:

1. Greater than \((>)\)
2. Less than \((<)\)
3. Equals to \((=)\)

Using the above-shown symbols, we can compare two numbers of any type, such as natural numbers, whole numbers, integers and decimal numbers etc. Thus, comparing and exploring the differences between the numbers is called the comparison of numbers.

Rules for Comparison of Numbers

There are some specific rules which will help to compare the numbers. Some of them are listed below:

1. Numbers having different number of digits
2. Numbers having the same number of digits

Rule 1: Numbers With Different Number of Digits

In comparing numbers, the number with more digits is always a greater number among the given, and the number with fewer digits is always smaller.

Example:

In the given numbers \(9999, 55, 2, 333\) 

\(9999\) is the greatest number as it has more \((4)\) digits, and the number \(2\) is the smallest as it has only one digit.

Some conclusions can be made based on the given rule as follows:

1. The two-digit numbers are always greater than the one-digit numbers
2. The three-digit numbers are always greater than the two-digit numbers
3. The four-digit numbers are always greater than the three-digit numbers
4. The five-digit numbers are always greater than the four-digit numbers, and so on.

Rule 2: Numbers With the Same Number of Digits

When comparing numbers with the same digits, we compare them starting from extreme left-most digits. Thus, the number with a greater extreme left-most digit is the greater number among them.

Example:

Compare the numbers \(632563\) and \(524329.\)

Here, given two numbers has the same number \((6)\) of digits, comparing the left-most digits, \(6\) is greater than \(5.\)

So, \(632563\) is greater than \(524329.\)

If the left-most digits of the given numbers are equal, then we will compare the next digit towards the right and so on.

Example:

Compare the numbers \(5700234\) and \(5100821.\)

Here, given numbers have the same number of digits, and the left-most digit \((5)\) is the same. So, next, we need to compare the following number towards the right, such as \((7)\) and \((1).\) 

So, \(5700234\) is the greater number.

How to Compare Large Numbers?

As we know, for comparing the greater and smaller number, we have some mathematical symbols.

We can remember the signs with the help of “Aligator Trick.”

We know that the alligator’s mouth always eats the larger amount, which will help us understand the greater than or less than sign.

Greater Than

As you can see, the alligator’s mouth is open towards the left, the same as the greater-than sign, in which wide-open side faces towards the left.

Example:
In the numbers \(999999\) and \(123456,\) the alligator’s mouth places towards the left \((999999)\). So, mathematically, we can show it as \(999999>123456.\)

Less Than

Here, the alligator’s mouth faced towards the right, same as a less-than sign, in which wide-open side faces towards the right.

Example:
In the numbers \(123456\) and \(999999\), the alligator’s mouth places towards the right \((999999)\). So, mathematically, we can show it as \(123456<999999\)

Importance of Comparing Large Number

Comparing numbers is an important part of building the number sense for the students. Number sense is the ability for a kid to recognize a number, its value, and its relationship with other numbers. 

Solved Examples – Comparing large numbers 

Q.1. Consider the least seven-digit number and the greatest six-digit number. Show the greater number by using the mathematical symbol.
Ans: We know that the least seven-digit number is \(1000000.\)
The greatest six-digit number is \(999999.\)
We know that the numbers with more digits are considered to be greater numbers.
So, \(1000000\) is the greater number.
In mathematical form, it can be written as follows:
So, \(1000000>999999.\)

Q.2. Put the correct sign \((<, =, >)\) for the following:
\(12345\)_________\(12333\)
\(11111111\)________\(11111111\)
\(5555555\)__________\(10000000\)
Ans: We know that to show the greater number, we can use the symbol \(”>”,\) and for a smaller number, we can use \(”>”.\)
So,
\(12345>12333\)
\(11111111=11111111\)
\(5555555<10000000\)

Q.3. Comparer the given numbers 87777766 and 88888888. Show the greater number among by using the Mathematical symbol.
Ans: Given,
\(87777766\) and \(88888888\)
We can see that the first digit is the same \(8=8.\)
So, we will compare the second digit from the left side \(7<8\)
So, \(88888888\) is the greater number.
In Mathematical form, it can be written as follows:
Hence, \(87777766<88888888.\)

Q.4. Comparer the given numbers 945326 and 865321. Show the greater number between them by using the mathematical symbol.
Ans: Given numbers are \(945326\) and \(865321.\) 
Here, we see that both the numbers have 6 digits. So we shall compare the digits from the left side one after the other.
We obtain \(9>8.\)
So, \(945326\) is the greater number.
In mathematical form, it can be written as follows:
Hence, \(945326>865321\)

Q 5. Comparer the given numbers 342654762 and 3476543218. Show the greater number using the mathematical symbol.
Ans: Given,
\(342654762\) and \(3476543218\)
We know that the numbers with more digits are considered to be greater numbers.
So, \(3476543218\) is the greater number as it has more digits compared to \(342654762.\)
In mathematical form, it can be written as \(342654762<3476543218.\)

Summary 

In this article, we have covered the large numbers, starting from what exactly numbers are, and then jumped on what is the comparison of large numbers followed by rules of comparing large numbers. After that, you glanced at how to compare the large numbers. Finally, we covered the importance of comparing large numbers and some solved examples and a few FAQs.

Frequently Asked Questions (FAQ) – Comparing Large Numbers

Q.1. How do you remember the mathematical symbols that are used for comparing large numbers?
Ans: As we know, for comparing the greater and smaller number, we have some mathematical symbols \((<, =, >)\)
We can remember the signs with the help of “Aligator Trick.”
We know that the alligator’s mouth always eats the larger amount, which will help us understand the greater than or less than sign.
Greater Than: Here, the mouth of the alligator open towards the left, the same as the greater-than sign, in which wide-open sides faces towards the left.
Less Than: Here, the alligator’s mouth faced towards the right, same as a less-than sign, in which wide-open side faces towards the right.

Q.2. How do you compare whole numbers with digits?
Ans: When one number is greater than another number, it may not be easy to compare the numbers effectively with a number line. 
Generally, whole numbers with more digits are greater than the whole numbers with less number of digits. 
For example, \(532\) is greater than \(94\) because \(532\) has three digits, and three-digit is always greater than two-digit.

Q.3. What is used to compare numbers?
Ans: We use the mathematical symbols to compare any two numbers, 
\(>:\) When one number is greater than the other number, we use the \(>\) greater than symbol to show the comparison.
\(=:\) When two numbers have the same value, we use the \(=\) equal to symbol to show the comparison.
\(<:\) When one number is smaller than the other number, we use the \(<\) smaller than symbol to show the comparison.

Q.4. How do you compare four-digit numbers?
Ans: The four-digit numbers are written or read according to their place values. In four-digit numbers, the four digits corresponded to the four-place values and they are ones, tens, hundreds and thousands. 
So, first, we start comparing from the thousands place if they are similar then move to hundreds place and so on.

Q.5. How do you compare two-digit numbers?
Ans: The two-digit numbers are written or read according to their place values. In two-digit numbers, the two digits corresponded to the two-place values and they are ones and tens. 
So, first, we start comparing from the tens place if they are similar then move to the ones place.

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