Expressing Large Number in Standard Form

As we study numbers, many large numbers become difficult to read. Expressing Large Numbers in Standard Form makes it easier. The standard form of numbers is introduced for decimal numbers, equations, polynomials, linear equations, etc. 

The exact definition can be explained better using decimal numbers and applying specific rules. You know that decimal numbers are the simplified form of fractions. This article will discuss how to express large numbers in standard form.

What is the Standard Form of a Number?

It is not easy to read numbers like \(12983678900000\) or \(0.0000000021982678.\) So when you want to read large and small numbers, you have to write the numbers in the standard form.

Definition: Any number that you can write as the decimal number, between the numbers \(1.0\) and \(10.0\), then multiplied by the power of the number \(10\), is known to be in the standard form.

Examples: Write the number \(81900000000000\) in the standard form:

\(81900000000000 = 8.19 \times {10^{13}}\)

Now, the number is written as \(10^{13}\) as we have moved the decimal point \(13\) places to the left side to get the standard form of the number \(8.19.\)

How do you write in Standard form?

The meaning of the standard form depends on the country you live in.

The standard form is another scientific notation in the United Kingdom and the countries using UK conventions.

\(3890 = 3.89 \times {10^3}\)

In the United States and the countries using the US conventions, the standard form is the usual way of writing numbers in the decimal notation.

The standard form of the number \(= 3890\)

The expanded form of the above number is \(3000 + 800 +90.\)

The word form of the above number is three thousand eight hundred and ninety.

Standard Form of a Rational Number

When we observe the rational numbers \(\frac{3}{5},\,\frac{{ – 5}}{8},\,\frac{2}{7},\,\frac{{ – 7}}{{11}}\)

The denominators of those rational numbers are positive integers, and \(1\) is the only common factor between the numerators and denominators. Further, the negative sign occurs only within the numerator.

These rational numbers are said to be in standard form.

Definition: A rational number is claimed to be within the standard form if its denominator is a positive integer and the numerator and denominator don’t have any common factor other than \(1\).

If you remember the method of reducing the fractions to their lowest forms, we divide the numerator and the denominator of the fraction by the same non-zero positive integer. We shall use an equivalent method for reducing the rational numbers to their standard form.

Example: Reduce to standard form \(\frac{{36}}{{ – 24}}.\)

Solution: So, the HCF of the numbers \(36\) and \(24\) is \(12\).

So, the standard form can be obtained by dividing the given fraction by \(-12\).

\(\frac{{36}}{{ – 24}} = \frac{{36 \div \left( { – 12} \right)}}{{ – 24 \div \left( { – 12} \right)}} = \frac{{ – 3}}{2}.\)

Expressing very large numbers in Standard Form

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

Example: \(5 × 5 × 5\) 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.

Similarly,

\((-2) × (-2) × (-2) × (-2)\) is written as \((-2)^4\)

\(\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3}\) is written as

\({\left( {\frac{2}{3}} \right)^{5}}\)

We have,

\(5 × 5 × 5 = 125\)

Also, \(5 × 5 × 5\) is written as \(5^3\).

\(\therefore 125 = {5^3}\)

Similarly,

\((-2) × (-2) × (-2) × (-2) = 16\)

\(⇒ (-2)^4 = 16\)

\(5^3\) is known as the exponential form or power notation of \(125\). Here, \(5\) is the base, and \(3\) is the exponent.

Similarly, \(16\) can be expressed as \((-2)^4\). Or the power notation of \(16\) with the base \(-2\) and the exponent \(4\).

We have,

\(100 = 10 × 10\)

\(1000 = 10 × 10 × 10\)

\(10000 = 10 × 10 × 10 × 10\)

\(100000 = 10 × 10 × 10 × 10 × 10\)

So, the exponential forms of \(100,\,1000,\,10000\), and \(100000\) are as given below:

\(100 = 10^2\) (read as \(10\) raised to the power \(2\))

\(1000 = 10^3\) (read as \(10\) to the power \(3\))

\(10000 = 10^4\) (read as \(10\) to the power \(4\))

\(100000 = 10^5\) (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’

Similarly, \(10^2\) is read as ‘\(10\) squared’ and \(10^3\) is read as ‘\(10\) cubed’.

It follows from the above discussion that for any rational number \(a\), we have

\(a × a = a^2\) (read as ‘\(a\) squared’ or ‘\(a\) raised to the power \(2\)’)

\(a × a × a = a^3\) (read as ‘\(a\) cubed’ or ‘\(a\) raised to the power \(3\)’)

\(a × a × a × a = a^4\) (read as \(a\) raised to the power \(4\) or the \(4^{\rm{th}}\) power of \(a\))

\(a × a × a × a × a = a^5\) (read as \(a\) raised to the power \(5\) or the \(5^{\rm{th}}\) power of \(a\)), and so on.

In general, if \(n\) is a natural number, then \(a × a × a × a …….× a = a^n\), \(n-\)times

\(a^n\) is called the \(n^{\rm{th}}\) power of a and is also read as ‘\(a\) raised to the power \(n\)’.

Note 1: It is evident from the above discussion that

\(a × a × a × b × b\) is written as \(a^3 b^2\) (read as \(a\) cubed into \(b\) squared)

\(a × a × b × b × b × b\) is written as \(a^2 b^4\) (read as \(a\) squared multiplied by \(b\) raised to the power \(4\))

Note 2: For any non-zero rational number \(a\), we define

\(a^1 = a\)

\(a^0 = 1\)

Example: Identify the value of each of the given numbers: \(11^2,\,9^3,\,5^4\)
We have, \(11^2 = 11 × 11 = 121\)
For \(9^3 = 9 × 9 × 9 = (9 × 9) × 9 = 81 × 9 = 729\)
In case of \(5^4 = 5 × 5 × 5 × 5 = 25 × 5 × 5 = (25 × 5) × 5 = 125 × 5 = 625.\)

Solved Examples

Q.1. Find the value of each of the following:
(i) \({\left( { – 3} \right)^2}\)
(ii) \({\left( { – 4} \right)^3}\)
(iii) \({\left( { – 5} \right)^4}\)
Ans: We have,
\({\left( { – 3} \right)^2} = \left( { – 3} \right) \times \left( { – 3} \right) = 9\)
We have,
\({\left( { – 4} \right)^3} = \left( { – 4} \right) \times \left( { – 4} \right) \times \left( { – 4} \right) = \left( { – 4} \right) \times \left( { – 4} \right) \times \left( { – 4} \right) = 16 \times \left( { – 4} \right) = – 64\)
We have,
\({\left( { – 5} \right)^4} = \left( { – 5} \right) \times \left( { – 5} \right) \times \left( { – 5} \right) \times \left( { – 5} \right)\)
\( = \left( {\left( { – 5} \right) \times \left( { – 5} \right)} \right) \times \left( { – 5} \right) \times \left( { – 5} \right)\)
\( = 25 \times \left( { – 5} \right) \times \left( { – 5} \right)\)
\( = \left( {25 \times \left( { – 5} \right)} \right) \times \left( { – 5} \right)\)
\( = \left( { – 125} \right) \times \left( { – 5} \right) = 125 \times 5 = 625.\)

Q 2. Simplify:
(i) \(2 \times {10^3}\)
(ii) \(7^2 \times {2^2}\)
(iii) \(2^3 \times {5}\)
(iv) \(0 \times {10^2}\)
Ans: We have,
\(2 \times {10^3} = 2 \times 1000 = 2000\)
In case of \({7^2} \times {2^2} = 49 \times 4\)
\(= 196\)
We have, \(2^3 \times 5\)
\(= 8 × 5 = 40\)
We have, \(0 × 10^2 = 0\) times \(100 = 0.\)

Q 3. Write the given number \(0.0000012\) in standard form:
Ans: Given, \(0.0000012\)
The standard form of the number 0.0000012 is \(1.2 × 10^{-6}\)
This is \(10^{-6}\) as we have moved the decimal point with six places to the right side to get the number \(1.2.\)

Q.4. Write the number \(62500000\) in the standard form.
Ans: Given, \(62500000\)
We will write the first six digits.
Then, by introducing the decimal, write the remaining digits except for the zeroes at the end \(6.25.\)
Now, you have to count the number of digits after the initial digit and then multiply \(10\) to the power of the number \(6.25 × 10^7.\)
Hence, the required answer is \(6.25 × 10^7.\)

Q 5. Multiply the numbers \(85000\) and \(2000\) and write the answer in standard form.
Ans: First, write the given numbers in standard form:
\(85000 = 8.5 \times {10^4}\)
\(2000 = 2 \times {10^3}\)
Now, multiply the numbers \(85000 \times 2000 = \left( {8.5 \times 2} \right) \times \left( {{{10}^{4 + 3}}} \right) = 17 \times {10^7} = 1.7 \times {10^8}\)
Hence, the required answer is \(1.7 \times {10^8}.\)

Summary

In the given article, we have discussed the standard form with an example. Then, we discussed how to write the numbers in standard form, followed by the standard form of rational numbers. We had glanced at the representation of decimal numbers and expressing very large numbers in standard form. You can also go through the solved examples provided with a few FAQs.

FAQs

Q.1. What is the standard form of large numbers?
Ans: 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: The given number \(5004300\) can be written in the standard form as shown below:
\(5004300 = 5.004300 \times {10^6}.\)

Q.2. How do you write \(450\) in standard form?
Ans: In general, the standard form of the number \(450\) is \(4.50 \times {10^2}.\)

Q 3. What is the standard form of \(12345\)?
Ans: The standard form of the number \(12345 = 1.2345 \times {10^4}.\)

Q.4. What is a standard form number?
Ans: Any number that you can write as the decimal number, between the numbers \(1.0\) and \(10.0\), then multiplied by the power of the number \(10\), is known to be in the standard form.
Examples: Write the number \(81900000000000\) in the standard form:
\(81900000000000 = 8.19 × 10^13\)
Now, the number is written as \(10^13\). We have moved the decimal point \(13\) places to the left side to get the standard form of the number \(8.19.\)

Q 5. How do you find standard form?
Ans: Observe the given steps to find the standard form of the number.
Example: Write the standard form of \(0.0009.\)
1. First, you have to write the number \(0.0009.\)
2. Next, you need to identify the decimal point in the given number. Then you can see the decimal point is after the four digits from the left side.
3. Now, you have to move the decimal point to the first non-zero digit in the given number.
4. Here, you will get \(9\), as there is no non-zero digit after the number \(9\), so there is no need to write the decimal point after \(9\).
5. You have to count the total number of digits you have moved the decimal point, which is four places further.
6. Now multiply the number with \(10\) and then raise the power of \(10\) with the total number of digits the decimal has been moved. The decimal point has been moved from left to right so that the power will be negative \( \to 9 \times {10^{ – 4}}.\)
7. Hence, the standard form of the number \(0.0009\) is \(9 \times {10^{ – 4}}.\)

We hope this article on expressing large numbers in standard form is helpful to you. If you have any queries on this page or in general about standard form numbers, ping us through the comment box below and we will get back to you as soon as possible.