# Standard Form in Math

**Decimal numbers in standard form** are the simplest form, that can be easily read and written. For example, the standard form of decimal number 0.007 is 7 × 10^{-3}. Sometimes it is very difficult to read or write very large or very small numbers. Hence, we write them in standard form. Not only decimals but any number can be written in standard form. In this article, we are going to learn what is the standard form of a number, how to express decimal numbers in standard form with many solved examples in detail.

## What is the Standard Form of a Number?

**A standard form of a number** in Maths is basically mentioned for the representation of large numbers or small numbers. We use exponents to represent such numbers in standard form.

The correct definition of standard form could be explained better in terms of decimal numbers and following certain rules. As we know, decimal numbers are the simplified form of fractions.

Some fractions give decimal numbers which have numbers after decimal at thousandths, hundredths or tenths place. But there are some fractions, which give a big decimal number. To represent such big numbers, we use simpler forms, which are also stated as Scientific notation.

Basically, we can say, it is the representation of rational numbers in standard form. Rational numbers are numbers that can be represented in the form of p/q, where p and q are both integers. For example, 1/10, 4/5, 8/9, etc.

## How to Write the Standard Form of a Number?

The steps to write the standard form of a number are as follows:

**Step 1:** Write the first number from the given number.

**Step 2:** Add the decimal point after the first number.

**Step 3:** Now, count the number of digits after the first number from the given number and write it in the power of 10.

For example, 52300000000 is the number. Thus the standard form of a number is obtained as follows:

**Step 1:** The first number is 5

**Step 2:** Adding the decimal point after 5, it becomes “5.”

**Step 3:** The number of digits after 5 is 10.

Hence, the standard form of 52300000000 is 5.23× 10^{10}.

## How to Write Decimal Numbers in Standard Form?

Decimal numbers employ 10 as the base and require 10 different numerals and a dot for the representation of its numbers. In this system, the digits used in denoting the number take different place values depending upon their position. For example, the number 645.221 can be written as 6 hundreds 4 tens 5 ones 2 tenths 2 hundredth and 1 thousandth.

Now, let us discuss how to write the decimal numbers in standard form.

**Step 1:** Write the first non-zero digit from the given number

**Step 2:** Add the decimal point after the first non-zero digit

**Step 3:** Find the number of the decimal point that shifts from the given number and write it to the power of 10.

For example, 0.0000017 in standard form is as follows:

**Step 1:** In the given number, the first non-zero digit is 1

**Step 2:** Add decimal point after the number 1. Hence, it becomes “1.”

**Step 3:** Here, the decimal point shifts 6 places to the right.

Hence, the standard form of a number 0.0000017 is 1.7 ×10^{-6}

In this section, we shall learn about the representation of numbers in the expanded form.

## Numbers in Expanded Form

Let us consider a 6 digit number 657891. In the expanded form this number can be written as:

657891 = 6 × 100000 + 5 × 10000 + 7 × 1000 + 8 × 100 + 9 × 10 + 1

We can express this number in the exponential form as well, as we know that 10000=10^{4}, 10 = 10^{0 }and so on.

657891 = 6 × 10^{5} + 5 × 10^{4} + 7 × 10^{3} + 8 × 10^{2} + 9 × 10^{1} + 1 × 10^{0}

## Expanded Form of Decimal Numbers

Similar to whole numbers, we can write the decimal numbers in expanded form, also. To decimal into expanded form, we need to multiply digits with increasing exponents of 1/10 or 10^{-1}.

Let us understand with the help of an example.

Example: Write 0.986 in expanded form.

0.986 = 9 × 10^{-1} + 8 × 10^{-2} + 6 × 10^{-3}

= 9 × 1/10 + 8 × 1/100 + 6 × 1/1000

= 0.9 + 0.08 + 0.006

## Standard Form of a Rational Number

A rational number ‘p/q’ is said to be in the standard form if the denominator q is positive and both the integers a and b have no common divisor other than 1.

**Steps to convert a rational number into the standard form:**

- Write the given rational number.
- Check if the denominator is positive or negative. If it is negative, then we need to multiply both numerator and denominator by -1, to make the denominator positive.
- Now, find out the greatest common divisor (GCD) of numerator and denominator, which could be cancelled.
- Divide the numerator and denominator by GCD.
- The obtained number is the Standard form of the given rational number.

## How to Express Large Numbers in Standard Form?

Any number can be expressed as decimal numbers between 1 and 10 multiplied by a power of 10. This form of representation is termed as the standard form. We can understand this concept with the following examples.

500 = 5 × 10^{2}

567 = 5.67 × 10^{2}

56.78 = 5.678 × 10^{1}

The advancement in science and technology has introduced us to numbers as big as the diameter of the Earth and as small as the size of a human cell. In order to represent these numbers, we use the exponential form to make their reading and writing more convenient.

### Related Articles

## Solved Examples

**Example 1:** Represent 567.21 in exponential form.

567.21 = (5 × 100) + (6 × 10) + 7 + (2 × 1/10) + (1 × 1/100)

Writing it in the exponential form:

567.21 = 5 × 10^{2} + 6 × 10^{1} + 7 × 10^{0} + 2 × 10^{-1} + 1 × 10^{-2}

**Example 2**: Represent the distance between the Earth and the Sun in exponential form.

Solution: The distance of the Earth from the Sun is 1496000000 km.

Therefore,

1496000000Km = 1.496 × 10^{9} Km

**Example 3**: Express the size of blood cells in the standard form.

Solution: The average size of blood cells in a human body is 0.000015 m

Therefore,

0.000015 = 1.5 × 10^{-5} m

Thus, we can say that the numbers when written in the standard form are much easier to read, understand and compare than when they are written in their actual form.

### Practice Questions

- Write the decimal number, 899.57, in standard form.
- What is the standard form of decimal number 0.00034?
- How to write 454000 in standard form?
- Convert 9.8000 x 10
^{-3}in decimal number.

## Frequently Asked Questions on Standard form of a Number

### What is the standard form of a number?

### What is the standard form of decimal numbers?

### Are the standard form and standard notation of decimal numbers, same?

### How to convert a decimal number into standard form?

### Write 0.00001487 in standard form.

^{-5}

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