An introduction to standard forms explained with examples

In mathematics, we strive to express everything logically in order to capture the reader's attention. If we face a number like 10900000000 which is difficult to read and the probability of error in reading and writing increases. The standard form has a key rule for the development of very big and small data values. This article explains the definition of standard form numbers, as well as applications and examples with step-by-step solutions.

What is the Standard form?

The standard form of a number is defined as “Any number between 1 to 10 multiplied with some power of 10”. We write it as a x 10ⁿ where 0 < a < 10 and n is an integer. The standard form is also called scientific notation. We can also replace 10 with e which is called the exponential standard form. The number also changes in the engineering notation.

Needs for Standard form

We cannot count a large number of values, such as 300000000 m/s, which is the speed of light. To reduce such difficulties, we write such large values in a standard notation which is called scientific notation. Similarly for very small values such as 0.00000000000000248, we change this small value in the standard form which helps us to read and understand the values easily.

Engineering notation:

It is a kind of scientific notation where we just concentrate on the exponential of 10 which must be a number that is divisible by 3. The notation is a x 10ⁿ where n is an integer that is divisible by 3.

How to solve standard form problems?

To change the given number in standard form we have followed some rules.

• First, we must determine where the decimal point is in the given value.
• We place the digit after the 1 st non-zero digit.
• If the non-zero digit is on the left side, the decimal is moved right to left, and if it is on the right side, the decimal is moved left to right.
• Count how many digits we have moved the decimal.
• Write the digit/number on the exponential of 10.
• The exponential is negative while moving decimal from right to left, and positive when moving from left to right.

To understand the conversion of a number in the standard form with the help of the following examples.

Example: 1

Convert 0.0000081 to the standard form.

Solution:

We solve the solution in the standard form by following the given steps.

Step 1: Write the given number.

0.0000081

Step 2: Identify the location of the decimal point in the number. We can observe that the decimal is on the left side of the number after seven digits.

Step 3: Decimal will be moved left to right and placed after the first non-zero digit. It will be

8.1

Step 4: Count the number of digits that we have moved the decimal. And right then that number is in the exponential of 10. We move six digits from left to right.

10⁶

Step 5: We move the decimal from left to right that’s the way the power of 10 must be with a negative sign.

8.1 x 10^-6

So, the standard form of 0.0000081 is 8.1 x 10^-6.

The above example can also be solved by using a standard form calculator to get the result in a fraction of a second.

• Enter any number in a given section
• Press the convert or enter key to get an answer.

Example 2:

Find the standard form of 81900000.

Solution:

We solve the solution of the given number by following the steps.

Step 1: Write the given number.

81900000

Step 2: Determine the decimal point in the number. If the member does not contain a decimal, then the decimal is added at the end of the number.

Step 3: Now we move the decimal from right to left side. And place at the 1 st non-negative digit

8.19

Step 4: Count the number of digits we move the decimal from right to left and write the number in the power of 10.

10⁷

Step 5: we move the decimal from right to left so the exponential of 10 must be positive.

8.19 x 10⁷

So the standard form of 81900000 is 8.19 x 10⁷.

Example 3:

Change 34578.2 into the engineering notation.

Solution:

Step 1: Write the given value.

34578.2

Step 2: Change the multiplication form so that the exponential of 10 is divisible by 3.

34.5782 x 10³

The 34.5782 x 10³ is the engineering form where the exponential is 3 which is divisible by 3.

Applications of standard form

• We use the standard form to find the prefixes
• To find the length of the blood cells.
• To find the distance between the sun and earth or other planets.

Summary

In this article, we have learnt about the need for standard form, definition, examples, and application. After this article, you are able to convert any large and small value into a notation that is easy to read and helps to reduce human mistakes which often occur in the reading.