# Standard Form in Mathematics: A Comprehensive Guide

In mathematics, a standard form is a way of representing larger and smaller experimental and distance numbers in a proper way. The standard form of numbers is a very easy approach to compare and simplify the arithmetic operations on two or more larger numbers or smaller numbers.

The guide of standard form will be its definition, notation, types, and calculations. You have to follow the basic guidelines of the standard form to master it. The representation of numbers could be done with the help of a well-known way.

## Standard Form in Mathematics

In mathematics, the way of representing larger or smaller numbers, distances, experiments, approximation values, and amounts in a concise and convenient way is known as standard form. It is also known as exponential form and scientific notation.

The term standard form is applicable when dealing with the distances in space such as planets, stars, and galaxies in which the numbers are extremely large. The microscopic and nanoscopic measurements through scales i.e., measuring the size of atoms and weight of molecules in which the numbers are extremely small.

### Notation of Standard Form

The standard form of smaller and larger observations can be dented as:

**K x 10 ^{y}**

In which K is the integral part of the number containing 1 to 9 integers and the term y is the power of the exponent. The power could be positive or negative.

- When we are dealing with larger numbers, the power of 10 would be positive.
- When we are dealing with smaller numbers, the power of 10 would be negative.

## Terms of Standard Form

There are various terms of standard form:

### A standard form of large numbers

The term standard form is very essential for dealing with larger numbers of a higher amount, distances, and volumes. For example, if the distance of a city is 2000000000000000 meters from the other city the standard form will be 2 x 10^{15}

The distance is very larger and difficult to read and write. You can write this number in scientific notation to concise it. Follow the below steps to write the numbers in scientific notation.

**Step 1:** Identify the coefficient.

First of all, take the first non-zero digit of the given number. The first non-zero digit is said to be the most significant digit of the larger number.

**Step 2:** Find the exponent.

Count the number of digits you moved to place the point at the first significant digit from right to left. The exponent will be positive in the case of larger numbers. After that write the number in the power of 10

**Step 3:** Write the number in the form of K x 10^{y}

Take the coefficient and the exponential part to write the equation in scientific notation.

### A standard form of small numbers

The term standard form is very essential for dealing with the smaller numbers of molecular weight, the size of atoms, and the measurements of smaller things. For example, if the size of an atom is 0.000000000000005 m it will be written in scientific notation as 5 x 10^{-14}

The size of an atom is very smaller and difficult to read and write. You can write this number in scientific notation to concise it. Follow the below steps to write the numbers in scientific notation.

**Step 1:** Identify the coefficient.

First of all, take the first non-zero digit of the given number. The first non-zero digit is said to be the most significant digit of the smaller number.

**Step 2:** Find the exponent.

Count the number of digits you moved to place the point at the first significant digit from left to right. The exponent will be negative in the case of smaller numbers. After that write the number in the power of 10

**Step 3:** Write the number in the form of K x 10^{y}

Take the coefficient and the exponential part to write the equation in scientific notation.

### A standard form of the rational number

The term standard form is very essential for dealing with the smaller and larger decimal numbers of rational numbers in the form of p/q. Follow the below steps to write the rational numbers in scientific notation.

**Step 1:** Convert the rational number to decimal

First of all, divide the numerator by the denominator and find the decimal number of the given rational number.

**Step 1:** Identify the coefficient.

Then take the first non-zero digit of the given number. The first non-zero digit is said to be the most significant digit of the decimal number.

**Step 2:** Find the exponent.

Count the number of digits you moved to place the point at the first significant digit from left to right or right to left. The exponent will be negative or positive in the case of smaller or larger numbers. After that write the number in the power of 10

**Step 3:** Write the number in the form of K x 10^{y}

Take the coefficient and the exponential part to write the equation in scientific notation.

## How to write numbers in scientific notation?

A standard form calculator is an easy to use tool to write various numbers in scientific notation without any effort. Here we are going to take a few examples to convert larger, smaller, and rational numbers in scientific notation manually.

### Example 1: For a Larger Number

Convert the given number 12357000000000 in scientific notation.

**Solution**

**Step 1:** Write the coefficient (1^{st} significant digit) of the given number.

take the first non-zero digit of the given number. The first non-zero digit is 6.

1.2357000000000

**Step 2:** Now determine the exponent.

Count the number of digits you moved to place the point at the first significant digit from right to left. The exponent will be positive in the case of larger numbers.

Exponent = 10^{13}

**Step 3:** Now write the number in scientific notation in the form of k x 10^{q}

k = 1.2357000000000

q = 10^{13}

Then the number would be:

k x 10^{q} = 1.2357000000000 x 10^{13}

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

1.2357 x 10^{13}

### Example 2: For Smaller Numbers

Find the scientific notation of 0.00000000002345

**Solution**

**Step 1:** Write the coefficient (1^{st} significant digit) of the given number.

take the first non-zero digit of the given number. The first non-zero digit is 2.

000000000002.345

**Step 2:** Now determine the exponent.

Count the number of digits you moved to place the point at the first significant digit from left to right. The exponent will be negative in the case of smaller numbers.

Exponent = 10^{-11}

**Step 3:** Now write the number in scientific notation in the form of k x 10^{q}

k = 000000000002.345

q = 10^{-11}

Then the number would be:

k x 10^{q} = 000000000002.345 x 10^{-11}

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

2.345 x 10^{-11}

### Example 3: For Rational Number

Find the scientific notation of 12/61

**Solution**

**Step 1: **Convert the given rational number into decimal form.

12/61 = 0.1967

**Step 2:** Write the coefficient (1^{st} significant digit) of the given number.

take the first non-zero digit of the given number. The first non-zero digit is 1.

01.967

**Step 2:** Now determine the exponent.

Count the number of digits you moved to place the point at the first significant digit from left to right. The exponent will be negative in the case of smaller numbers.

Exponent = 10^{-1}

**Step 3:** Now write the number in scientific notation in the form of k x 10^{q}

k = 01.967

q = 10^{-1}

Then the number would be:

k x 10^{q} = 01.967 x 10^{-1}

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

1.967 x 10^{-1}

## Conclusion

Now you can grab the basics of the term standard form from the above comprehensive guide. You can take assistance from this post to write larger, smaller, and rational numbers in scientific notation.