Standard Form of Numbers: Explanation and Examples with Solutions

In mathematics, the standard form of numbers is an essential technique to denote the larger and smaller numbers in an efficient way. When we dealing with the very large numbers and very small numbers, then this branch of mathematics is used.

It is frequently used in different fields of science, engineering, and mathematics. The standard form is very effective way to write numbers in shorter and easy to write way. In this article, we are going to explore standard form of numbers with the help of examples with solutions.

What is the standard form of numbers?

In number theory, the term standard form is a way of representing larger and smaller numbers in a standardized format. This technique of representing numbers is also known as the scientific notation of numbers.

This term is used to write very lager numbers and very small numbers in a manageable and easy to write and read manner. It follows the below format to write numbers in standard form.

P x 10^Q

Where

• P is a decimal number between 1 and 10
• Q is an integer representing the exponent of 10

Numbers written in this format allow us to convey the magnitude of a value and make calculations with large and small numbers more efficient. Here are a few reasons why it is widely used:

Dealing with the Large Numbers

Large numbers are involved in many solar and scientific calculations that could make the representation of regular decimal form in a bulky way. Then dealing with such large number of calculations, the term standard form is used to represent them in a concise and readable manner.

It is more convenient to express distances between astronomical bodies in standard form, such as stars and galaxies. For example, the distance of the sun from the Earth is 150000000000 meter that is a large number. We can write it in standard form such as 1.5 x 10¹¹.

Dealing with the Small Numbers

Small numbers are involved in many chemical calculations and size of the objects that are difficult to see with the naked eye. Such as the size of atoms and wavelength of electromagnetic waves.

The calculations of the smaller objects make the regular decimal notations very complicated due to the presence of the number of zeros in the smaller number. Then dealing with such small number of calculations, the term standard form is used to represent them in a concise and readable manner.

How to Convert Numbers into Standard Form?

Here we are going to explore some steps to convert the numbers in the standard form.

For Larger Number

Let’s the number is 4565000000000

Step 1: Write the given number in decimal form.

Write the given number in the decimal notation by placing the decimal point after the first significant digit.

4.565000000000

Step 2: Representing the decimal point with an exponent.

Identify the position of decimal number and count the digits after the decimal point from right to left. And write them in the power of 10 with the positive sign.

10¹²

Step 3: Then the number should be written in according to the general format P x 10^Q.

P = 4.565000000000

Q = 10¹²

Then the number would be:

4.565000000000 x 10¹²

Step 4: The zeros at the right should be ignored.

4.565 x 10¹²

For Smaller Number

Let’s the number is 0.00000000565

Step 1: Write the given number in decimal form.

Write the given number in the decimal notation by placing the decimal point after the first significant digit.

0000000005.65

Step 2: Representing the decimal point with an exponent.

Identify the position of decimal number and count the digits after the decimal point from left to right. And write them in the power of 10 with the negative sign.

10^-9

Step 3: Then the number should be written in according to the general format P x 10^-Q.

P = 0000000005.65

Q = 10^-9

Then the number would be:

0000000005.65 x 10^-9

Step 4: The zeros at the left should be ignored.

5.65 x 10^-9

A standard form of numbers calculator can also use to find the standard form of larger and smaller numbers in seconds. Below is a screenshot of the above example solve through this tool:

Arithmetic Operations with Standard Form

Now we are going to explore how to perform arithmetic operations such as addition, subtraction, multiplication, and division with standard form numbers.

Addition and Subtraction in Standard Form

It is important to ensure that the exponents of 10 are the same when performing the addition and subtraction in standard form, Here’s the step-by-step process:

For addition

1. Align the Exponents: Align the numbers by adjusting the coefficients and the exponents to have the same exponent of 10.
2. Perform the Calculation: Add the coefficients while keeping the exponent of 10 unchanged.
3. Adjust the Result: If the result is not in standard form, convert it back to standard form by adjusting the coefficient and exponent.

For subtraction

1. Align the Exponents: Align the numbers by adjusting the coefficients and the exponents to have the same exponent of 10.
2. Perform the Calculation: Subtract the coefficients while keeping the exponent of 10 unchanged.
3. Adjust the Result: If the result is not in standard form, convert it back to standard form by adjusting the coefficient and exponent.

For example

Add and subtract the given number.

3.56 x 10¹² + 51.3 x 10¹¹

Solution

Step 1: Align the numbers by adjusting the coefficients and the exponents to have the same exponent of 10.

The second number has less exponent, we have to align the second number to make the similar power.

3.56 x 10¹² + 5.13 x 10¹¹⁺¹

3.56 x 10¹² + 5.13 x 10¹²

Step 2: Add the coefficients while keeping the exponent of 10 unchanged.

 (3.56 + 5.13) x 10¹²

8.69 x 10¹²

Step 3: The coefficient and exponent should be adjusted if the result isn’t standard form.

The calculated result is in standard form

For example

Subtract the given number.

35 x 10¹² – 150 x 10¹¹

Solution

Step 1: Align the numbers by adjusting the coefficients and the exponents to have the same exponent of 10.

The second number has less exponent, we have to align the second number to make the similar power.

35 x 10¹² – 15.0 x 10^{11 + 1}

35 x 10¹² – 15.0 x 10¹²

Step 2: Subtract the coefficients while keeping the exponent of 10 unchanged.

 (35 – 15.0) x 10¹²

20 x 10¹²

Step 3: The coefficient and exponent should be adjusted if the result isn’t standard form.

2.0 x 10^{12 + 1}

2.0 x 10¹³

Multiplication and Division in Standard Form

For multiplication

1. Multiply the Coefficients: Multiply the coefficients of the numbers while keeping the same exponent of 10
2. Add the Exponents: Add the exponents when multiplying
3. Adjust the Result: If necessary, convert the result back to standard form by adjusting the coefficient and exponent.

For Division

1. Divide the Coefficients: Divide the coefficients of the numbers while keeping the same exponent of 10
2. Subtract the Exponents: Subtract the exponents when dividing.
3. Adjust the Result: If necessary, convert the result back to standard form by adjusting the coefficient and exponent.

For example

Multiply the given number.

3 x 10¹⁰ x 5 x 10¹⁴

Solution

Step 1: Multiply the coefficients of the numbers while keeping the same exponent of 10

(3 x 5) x 10¹⁰ x 10¹⁴

15 x 10¹⁰ x 10¹⁴

Step 2: Add the exponents

15 x 10^{10 + 14}

15 x 10²⁴

Step 3: Convert the result back to standard form by adjusting the coefficient and exponent.

1.5 x 10^{24 + 1}

1.5 x 10²⁵

For example

Divide the given number.

80 x 10²⁰ / 2 x 10¹⁴

Solution

Step 1: Divide the coefficients of the numbers while keeping the same exponent of 10

(80 / 2) x 10²⁰ / 10¹⁴

40 x 10²⁰ / 10¹⁴

Step 2: Subtract the exponents

40 x 10^{20 – 14}

40 x 10⁶

Step 3: Convert the result back to standard form by adjusting the coefficient and exponent.

4.0 x 10^{6 + 1}

4.0 x 10⁷

Final Words

Now you can take assistance from this post to convert the larger and smaller numbers into the standard from. You can get the step by step guidance for the conversion of the numbers. You can also get info about performing the arithmetic operations on standard form numbers.