 # Standard Form In Mathematics

This article will give you a detailed guide on how to express numbers in a unique form known as the standard form. We are going to illustrate this topic using several worked examples to ease understanding. Ensure to read this article to the end.

## What Is Standard Form?

Standard form is a unique way of expressing very large or very small numbers e.g 1,222,333,645 or 0.0009898 using a shorter and easily written format. It is usually expressed mathematically as:

a x 10n

Where;

a is a positive or negative number between 1 and 9

n is a positive or negative integer

The standard form makes numbers that were otherwise very large to write down or compute, small and easier to use. It makes use of an index (powers) to express numbers in an easily understood format.

## Step-by-step Guidelines For Converting Numbers To Standard Form

We shall use some worked examples to show you how to convert ordinary numbers to standard forms.

### Example

Write the following numbers in their standard form:

• 5187362000
• 345674
• 420.46
• 00000008776897
• 002389978
• 23.34

#### Solution

• 5187362000

Step one: Check the nature of the number that is, whether it is a positive or negative number. In our case, 5187362000 is a positive number.

Step Two: Add a decimal point after the first integer in the number. In our case, we shall add a decimal point after 5

5.187362000

Step Three: Recall that the general formula for standard form is a x 10n. Count the number of digits (including 0) after the decimal point. The number of digits serves as the power or index of the standard form. In this question, our index is 109

Step Four: Write out the non-zero digits and then multiply by the index derived above.

= 5.187362 x 109

• 345674

We are going to repeat all the steps above.

3.45674

Number of digits after the decimal point = 5

= 3.4 x 105

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Changing The Subject Of The Formula

• 420.46

This question poses an unusual problem; it comes with a decimal point already. To solve this easily, you have to recall your number line.

As a positive number, you are required to express the standard form in one decimal place. Therefore, to solve this, you will count the number of places you moved back to get the first decimal point. That will serve as your index

420.46

= 4.2046 x 102

• 00000008776897

To solve this, you have to follow these steps

Step one: Move the decimal point to the first non-zero digit

Step two: count the number of 0 before the first non-zero digit, the number is the index. In this case, the number is 8. However, you will express the index as 10-8 to indicate that there are zeros ahead of the number.

This is so because; 0.00000008776897 = 8.776897/100000000

Step three: write out the full number

= 8.776897 x 10-8

You can approximate it to two decimal places

= 8.78 x10-8

• 002389978

= 2.39 x 10-3

• 23.34

= 2.334 x 101

## Converting From Standard Form To Ordinary Form

You can also convert from standard form to ordinary numbers.

Example: convert 4.2 x 102 and 3.4 x 10-3 to ordinary form

### Solution

(i) 102 = 10 x 10 = 100

4.2 x 100 = 420

(ii) 10-3 = 1/10 x 1/10 x 1/10 = 0.001

3.4 x 0.001 = 0.0034

## Things To Note About Standard Form

• Any number can be converted into standard form regardless of whether they are very large or very small. In as much as you know and follow the rules of conversion accordingly, you will convert it.
• In the standard form general formula, a can either be a whole number or a decimal fraction. Also, n can be zero.
• You need to have a sound knowledge of the number line.
• A good knowledge of the power law of indices is needed to easily solve arithmetic problems involving this form.
• Standard form is very important to astronomers as their field requires the computation of very large numbers as well as very small numbers.

This article has clearly shown you how to express numbers in their standard forms. If you have further questions, comments, or inquiries about this article, endeavor to use the comment box below. Also, ensure to subscribe to this blog for more interesting educational guides like this.