Numbers are a basic element in our daily life. Numbers may be challenging, especially when there are several digits involved. To make such numbers easier to understand, mathematicians use the standard form method. The standard form of a number helps to write extremely large or extremely small numbers easier.

In this article, we will talk about the standard form of a number with its mathematical representation. We will learn how to express decimals numbers and rational numbers in standard forms. We will solve some examples related to the standard form of the number.

## Definition, and mathematical representation of the standard form

The standard form describes very large or very small numbers more concisely. It is often known as scientific notation. In standard form, we express a number as a multiple of the decimal number between 1 and 10 with the exponent of 10.

In other words, a standard form is a number written as b × 10^{n}, where 1 ≤ b < 10 and n is an integer. In not only mathematics but also it is widely used in science. For example, the mass of the proton in standard form can be written as 1.673 × 10^{-27}.

**Representation of whole numbers in standard form**

All non-negative integers including zero are considered whole numbers, with no negative number or fractions included. Follow these steps to write whole numbers in standard form:

- Write the first digit from the left of the given number.
- Place the decimal point after the first given digit.
- Determine the exponent of 10 by counting the number of digits after the first digit.

For example, the whole number 65,000,000,000 can be written in standard form as:

**Step 1****:** Write the first digit from the left of the given number, that is 6.

**Step 2****:** Place the decimal point after the first given digit. , i.e. “6.”

**Step 3****:** 10 numbers are here after the first digits.

The standard form of the whole number 65,000,000,000 is 6.5× 10^{10}.

## Steps for writing the decimal numbers in standard form

Follow the steps to write the decimal numbers in standard form:

- Write down the first non-zero digit of the given number.
- After the first non-zero digit, place the decimal point.

- Determine the number of times the decimal point need to move to get the original number.

- Count how many times the decimal point moves to the original value. If the decimal point moves from the right to the left, write a negative sign on the exponent of 10.
- Write the determined number of the decimal shifts as a power of 10.

The number decimal number 0.00000000000057 can be expressed in standard form by the following method.

**Step 1****: **Here, 5 is the first non-zero digit of the given number.

**Step 2****: **Put the decimal point after 5, i.e. “5.”

**Step 3****:** As the decimal point moved 10 places to the right side. So write this number in the power of 10.

The standard form of 0.00000000057 is 5.7 × 10^{-10}.

## Steps to write the expanded form in a standard form

The standard form of the expanded form can be obtained by adding all given values or writing the only first digits of all given values.

For examples: 4000000 + 300000 + 00000 + 2000 + 100 + 20 + 2

Here, 4, 3, 0, 2, 1, 2, and 2 are the first digits of the given values. Hence, the standard form of the given expanded form is 4302122.

## How to write the Rational numbers in their standard form

The term “Rational numbers” refers to numbers that can be expressed in the form of p / q (where q ≠ 0). If the greatest common divisor of p and q is 1 then p / q will be said standard form of rational number. For example, the standard form of 6 / 4 is 3 / 2, because the greatest common divisor of 3 and 2 is 1.

## Examples related to Standard forms of the number

We will learn how to ordinary numbers will convert to standard form with the help of some examples.

**Example1.**

Write the standard form of the following whole number.

7,594,002,000.

**Solution **

Write the first digit from the left of the given number, which is 7.

Place the decimal point after seven, i.e. “7.”

Here are 9 digits after the first 7.

Write that number to the exponent of 10. i.e. 10^{9}

Putting it all together, the standard form of the whole number 7,594,002,000 is 7.594002 × 10^{9}

**Example 2.**

Write the decimal number 0.000005056 in standard form.

**Solution **

Write the first digit from the left, which is 5

Place the decimal number after five. i.e. “5.”

Write the determined number of the decimal shifts as a power of 10. I.e. 10^{-6} (As we shifted the decimal point 6 places to the right)

Putting it all together, the standard form of the given decimal number 0.000005056 is 5.056 × 10^{-6}.

**Example 3.**

Find the standard form of the following expanded form

90000000 + 5000000 + 000000 + 40000 + 3000 + 200 + 10 + 9

**Solution **

Simplify the given expression by adding the numbers together:

90000000 + 5000000 + 000000 + 40000 + 3000 + 200 + 10 + 9 = 95,043,219

The standard form of the given expanded from 95,043,219.

## Conclusion

In this article, we have discussed the standard form of the number with its mathematical expression. We described how to express decimal numbers and rational numbers in the standard form. We discussed how to write the standard form of the expanded form.

Further, we did solve different examples of the standard form of the number. After understanding this article, you will be able to write the standard form of decimal numbers and rational numbers.

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