Comparing Very Large and Very Small Numbers: Definition

Comparing Very Large and Very Small Numbers: It is not very convenient to read, understand and compare very large and very small numbers. To make such large and small numbers easy to read, understand and compare, we first need to convert them into their standard form using exponents and then compare. Some examples of very large numbers are \(89,000,000,000\) and \(5,978,043,000,000,000\) and very small numbers are \(0.345623467\) and \(0.00000000000047845\).

We can also compare very large and very small numbers by obtaining their ratios. In this article, we shall discuss how to compare large and small numbers with some solved examples.

Large Numbers in Standard Form Using Exponents

Exponents: We know that the continued sum of a number added to itself several times can be written as the product of a natural number, equal to the number of times it is added and the number itself.

Example: \(5+5+5+5+5+5+5=7×5\)
\((-2)+(-2)+(-2)+(-2)+(-2)+(-2)+(-2)+(-2)+(-2)=9 \times(-2)\)
\(\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}=5 \times \frac{3}{4}\)
\(0.3+0.3+0.3+0.3+0.3+0.3=6 \times 0.3\) etc.

Similarly, the continued product of a number multiplied by itself several times can be written as the number raised to the power a natural number, equal to the number of times the number is multiplied with itself.

Thus, \(5×5×5\) can be written as \(5^{3}\) and it is read as \(5\) raised to the power \(3\) or third power of \(5\). In \(5^{3}\), we call \(5\) as the base and \(3\) as the exponent.

More example:  We shall take the same set of numbers, but take their continued product and express them as follows:

\(5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5=5^{7}\)
\((-2) \times(-2) \times(-2) \times(-2) \times(-2) \times(-2) \times(-2) \times(-2) \times(-2)=(-2)^{9}\)
\(\frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4}=\left(\frac{3}{4}\right)^{5}\)
\(0.3 \times 0.3 \times 0.3 \times 0.3 \times 0.3 \times 0.3=(0.3)^{6}\)

In the same way, we can write, 

\(100=10 \times 10=10^{2}\) (read as \(10\) raised to the power \(2)\)
\(1000=10 \times 10 \times 10=10^{3}\) (read as \(10\) to the power \(3)\)
\(10000=10 \times 10 \times 10 \times 10=10^{4}\) (read as \(10\) to the power \(4)\)
\(100000=10 \times 10 \times 10 \times 10 \times 10=10^{5}\) (read as \(10\) to the power \(5)\)

Some powers have special names. For example:

\(5^{2}\) which is \(5\) raised to the power \(2,\) is also read as \(‘5\) squared’

\(5^{3}\) which is \(5\) raised to the power \(3,\) is also read as \(‘5\) cubed’ 

Standard Scientific Notation of Representing a Number

Scientific notation is just a different way of writing the same number in a standard way. Any number is considered represented in a standard scientific way if it is represented as \(M \times 10^{n}\) where \(M\) has only one digit among \(1, 2, 3, 4, 5, 6, 7, 8\) and \(9\) before the decimal point, and it may have one or more digits after the decimal point. n is an integer.

Example: 

\(2400=2400.0=2.4 \times 10^{3}\)
\(123456789=1.23456789 \times 10^{8}\)
\(0.0045678=4.5678 \times 10^{-3}\)
\(0.00000000000000000000723456789 = 7.2345678 \times 10^{-21}\)

Use of Symbols for Comparing Very Large & Small Numbers

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

How to Compare Very Large and Very Small Numbers?

We can compare very large and very small numbers using: 

a) Standard scientific notation of the numbers
b) Ratios

Compare Very Large Numbers Using Standard Scientific Notation

In this method, first, we compare the exponent part of the two numbers. The number having the larger exponent is greater.

Example-1: Distance of the Sun from the Earth \(=14960000000000 \mathrm{~km}=1.496 \times 10^{13} \mathrm{~km}\)

Distance of the Moon from the Earth \(=384440 \mathrm{~km}=3.8444 \times 10^{5} \mathrm{~km}\)

We observe that the distance of the Moon from the Earth is \(3.8444 \times 10^{5}\) has exponent \(5,\) but the distance of the Sun from the Earth is \(1.496 \times 10^{13}\) has exponent \(13.\)

Hence, we say that \(1.496 \times 10^{13} \mathrm{~km}\) is greater than \(3.8444 \times 10^{5} \mathrm{~km}\) and we write \(1.496 \times 10^{13} \mathrm{~km}>3.8444 \times 10^{5} \mathrm{~km}\)

If the exponents are the same, then we compare the \(M\) part of the number \(M \times 10^{n}\) in the usual way. (We compare the whole number part of \(M\) first. The number having the bigger whole number is bigger. If the whole number of parts are the same, we compare the decimal digits one after the other.)

Example-2: So we have, \(5.496 \times 10^{17}>1.496 \times 10^{17}, \quad 5.496 \times 10^{17}>5.396 \times 10^{17}\)

\(5.496 \times {10^{17}} < 5.497 \times {10^{17}}\) and so on.

Compare Very Small Numbers Using Standard Scientific Notation

In this method also, first, we compare the exponent part of the two numbers. The number having the larger exponent is greater.

Example-1: Let us consider the numbers \(6.435 \times 10^{-25}\) and \(2.747 \times 10^{-22}\)

Hence, we observe that the exponent \(-22\) in \(2.747 \times 10^{-22}\) is greater than the exponent \(-25\) in \(6.435 \times 10^{-25}\)

Hence, we say \(2.747 \times 10^{-22}>6.435 \times 10^{-25}\)

If the exponents are the same, then we compare the \(M\) part of the number \(M \times 10^{n}\) in the usual way. 

Example-2: So, we have \(5.496 \times {10^{ – 17}} > 1.496 \times {10^{ – 17}},\quad 5.496 \times {10^{ – 17}} > 5.396 \times {10^{ – 17}},\)

\(5.496 \times {10^{ – 17}} < 5.497 \times {10^{ – 17}}\) and so on.

Compare Very Large Numbers Using Ratio

In this method, first, we obtain the ratio of the two given numbers. If the ratio is greater than \(1,\) then the number in the numerator is greater than the number at the numerator.

Example-1: Distance of the Sun from the Earth \(=14960000000000 \mathrm{~km}=1.496 \times 10^{13} \mathrm{~km}\)

Distance of the Moon from the Earth \(=384440 \mathrm{~km}=3.8444 \times 10^{5} \mathrm{~km}\)

We have, \(\frac{{{\rm{ distance\, of\, the\, Sun\, from\, the\, Earth }}}}{{{\rm{ distance\, of\, the\, Moon\, from\, the\, Earth }}}}{\rm{ = }}\frac{{{\rm{1}}{\rm{.496 \times 1}}{{\rm{0}}^{{\rm{13}}}}}}{{{\rm{3}}{\rm{.8444 \times 1}}{{\rm{0}}^{\rm{5}}}}}{\rm{ = 3}}{\rm{.891 \times 1}}{{\rm{0}}^{\rm{7}}}{\rm{.}}\)

We observe that the ratio is larger than \(1.\) Hence, we say that the distance of the Sun from the Earth is greater than the distance of the Moon from the Earth.

We write \(1.496 \times 10^{13} \mathrm{~km}>3.8444 \times 10^{5} \mathrm{~km}\)

Example-2: Let us compare the numbers \(1.496 \times 10^{17}\) and \(5.496 \times 10^{17}\)

We have \(\frac{{1.496 \times {{10}^{17}}}}{{5.496 \times {{10}^{17}}}} = 0.27\)

We observe that the ratio is less than \(1.\) Hence, \(1.496 \times 10^{17}<5.496 \times 10^{17}\)

Compare Very Small Numbers Using Ratio

In this case, also we proceed the same way as comparing very large numbers using ratio. We obtain the ratio of the two given numbers. If the ratio is greater than \(1,\) then the number in the numerator is greater than the number at the numerator.

Example-1: Given that the diameter of a red blood cell \(=5 \times 10^{-8} \mathrm{~m}\) and that of a plant cell \(6.25 \times 10^{-11} \mathrm{~m}\) We shall compare their diameters.

Thus we have, \(\frac{{{\rm{red\, blood\, cell }}}}{{{\rm{ plant\, cell }}}}{\rm{ = }}\frac{{{\rm{5 \times 1}}{{\rm{0}}^{ – 8}}}}{{{\rm{6}}{\rm{.25 \times 1}}{{\rm{0}}^{{\rm{ – 11}}}}}}{\rm{ = 800}}\)

We observe that the ratio is larger than \(1.\) Hence, we say that the diameter of the red blood cell is greater than that of a plant cell. 

We write \(5 \times 10^{-8} \mathrm{~m}>6.25 \times 10^{-11} \mathrm{~m}\)

Example-2: Let us compare the numbers \(1.496 \times 10^{-17}\) and \(5.496 \times 10^{-17}\)

We have \(\frac{1.496 \times 10^{17}}{5.496 \times 10^{17}}=0.27\)

We observe that the ratio is less than \(1.\) Hence, \(1.496 \times 10^{-17}<5.496 \times 10^{-17}\)

Solved Examples

Q.1. Which one is greater \(12^{-13}\) or \(12^{-12}\)?
Ans: We observe that in the given two numbers, the base is the same \((12.)\) But exponent \(-12\) in \(12^{-12}\) is greater than the exponent \(-13\) in \(12^{-13}\) Therefore, \(12^{-12}\) is greater than the number \(12^{-13}\)

Q.2. Identify the greater number in the following:
\(2675^{36}\) and \(2675^{35}\)
Ans: \(2675^{36}\) is greater than the number \(2675^{35}\) because although the base of both the two numbers is the same, the number \(2675^{36}\) has a greater exponent.

Q.3. Which one is less \(4.657^{-32}\) or \(6.875^{-27}\)?
Ans: In the given two numbers \(4.657^{-32}\) and \(6.875^{-27}\) the exponent \(-27\) in \(6.875^{-27}\) is greater than the exponent \(-32\) in \(4.657^{-32}\) Hence, \(4.657^{-32}<6.875^{-27}\)

Q.4. Which one is greater \(2^{3}\) or \(3^{2}\)?
Ans: We have, \(2^{3}=2 \times 2 \times 2=8\)
\(3^{2}=3 \times 3=9\)
Since \(9>8,\) so, \(3^{2}\) is greater than \(2^{3}\)

Q.5. Which one is greater \(8^{2}\) or \(2^{8}\)?
Ans: We have
\(8^{2}=8 \times 8=64\)
\(2^{8}=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2=256\)
Clearly,\(2^{8}>8^{2}\)

Summary 

This article discussed the term very large and small numbers using a standard form, exponents, and examples. We discussed how a number is represented using the standard form. We have also discussed how to compare the very large numbers and very small numbers using a standard form and the ratio of two given numbers. We have given solved examples along with a few of the FAQs.

Frequently Asked Questions (FAQs)

Q.1. How do you compare very large and very small numbers?
Ans: We may compare very large and very small numbers by obtaining their ratios. If the ratio is more than \(1,\) then the number at the numerator is greater.
Comparing very large numbers:
Let us compare the numbers \(2.8 \times 10^{21}\) and \(5.6 \times 10^{22}\)
Their ratio is \(\frac{2.8 \times 10^{21}}{5.6 \times 10^{22}}=\frac{1}{20}\).
The ratio is less than \(1.\) Hence, the numerator \(2.8 \times 10^{21}\) is less than the number \(5.6 \times 10^{22}\)
Comparing very small numbers:
Let us compare the numbers \(8 \times 10^{-8}\) and \(7.25 \times 10^{-12}\)
Their ratio is \(\frac{8 \times 10^{-8}}{7.25 \times 10^{-12}}=\frac{80000}{7.25}=\frac{8000000}{725}=\frac{320000}{29}\)
The ratio is greater than \(1.\) Hence, the numerator \(8 \times 10^{-8}\) is greater than the number \(7.25 \times {10^{ – 12}}\)

Q.2. What is the large number and small number?
Ans: Large and small numbers are generally considered numbers bigger than or small than what is used in daily life. 
Example: Large numbers
\(27,000,000,000,\)
\(6,312,079,000,000,000\) 
Small numbers
\(0.12345678,\)
\(0.00000000000054638\)

Q.3. What is a way of writing down very large or very small numbers easily?
Ans: The easy way of writing down very large or very small numbers is to express the number in standard scientific notation.
A number is said to be standard is expressed as the product of a number between \(1\) and \(10\) (including \(1\) but excluding) and a positive integer power of \(10.\)
\(10^{3}=1000\), so \(4 \times 10^{3}=4000\). So \(4000\) can be written as \(4 \times 10^{3}\)
\(10^{-9}=0.000000001\) and \(10^{-6}=0.000001\)

Q.4. What is the highest countable number?
Ans: Googol. It is a large number, unimaginably large. It is easy to write in the exponential format: \(10^{100}\) an extremely compact method, to easily represent the largest numbers (and most minor numbers).

Q.5. What form do we use to write to very large numbers or very small numbers?
Ans: Standard form, or standard index form, is the writing number system that can be particularly helpful for working with very large or very small numbers. It is based on using powers of \(10\) to express how big or small a number is.