# Standard Form in Mathematics – An Easy Way to Express Numbers

Standard form in mathematics is a concise way to write extraordinarily large or extremely small numbers. It is particularly useful in various scientific and engineering fields where dealing with such numbers is common.

[Note: At this time of writing, the notation written originally in LaTeX using the MathML plugin displays correctly only in Microsoft Edge.]

The distance from the Earth to the Sun is approximately 93000000 miles, which can be expressed in standard form as 9.3 × 10^7. It allows for easier comparison and calculation of such numbers.

This article will help you grasp the concept of standard form for numbers. We will define it and show how to write numbers this way. We will solve many examples to make it easier to understand.

## Definition and Explanation of Standard Form

The standard form or scientific notation is a method of expressing a number as a coefficient multiplied by a power of 10. It is frequently used to write extremely large or very small numbers in a more compact and manageable form.

Numbers are written in standard form as follows:

$b\times 10^{n}$

Where:

• b is a coefficient greater than or equal to 1 and less than 10.
• n is an exponent that indicates the power of 10 by which b is multiplied or divided.

## Examples of Standard Form in Mathematics

Consider the number 45,600,000. A more convenient representation of this number can be expressed as

$4.56\times 10^{7}$

When you examine it closely, 4.56 falls within the range of decimal numbers from 1.0 to 10.0. Thus, we can represent 45,600,000 in standard form as 4.56 × 10^7.

### Standard Form of Whole Numbers

Here is how to express whole numbers in standard notation:

1. Take the first number from the given whole number and write it down.
2. Place a dot (decimal point) right after that first number.
3. Count the number of digits after the first number from the given number and express it as the power of 10.

#### Example:

Express the whole number 9500000 in standard form.

Step 1: Write the 1st number from the given number that is 9.

Step 2: Place the decimal point after writing the first digit ‘9.’

Step 3: Count the number of digits after the first number (6 digits). Write it as the power of 10.

Standard form:

$9.5\times 10^{6}$

### Decimal Numbers in the Standard Form

Representing decimal numbers in standard form also involves expressing a number as a product of a single digit (between 1 and 9) and a power of 10. Follow the following steps to represent the decimal number in standard notation.

1. Identify the first digit non-zero digit of the decimal point.
2. Write down this digit as the first number in standard form.
3. Place a decimal point immediately after the first non-zero digit.
4. Determine the number of decimal places the original decimal point needs to shift to reach its new position. Write this number as an exponent of 10.

Note: When the decimal point jumps to the left side, the power of 10 will be positive. The exponent of 10 is negative when the decimal point shifts right.

#### Example:

Convert 0.00567 to standard form.

Step 1: The first non-zero digit is 5.

Step 2: Write down 5.

Step 3: Put the decimal point after 5, so it will become ‘5.’

Step 4: Find the total count of digits that appear after the decimal point (3 digits in this case). Write it as a negative power of 10.

Standard Form:

$5.67\times 10^{-3}$

A standard form calculator is an easy to use tool to deal with the problems of writing greater and smaller numbers in standard notation.

## Calculation of Standard Form

Standard form numbers can be manipulated as follows:

• First, the powers of 10 should be made equal.
• Then, the coefficients are added or subtracted depending on the operation.
• Finally, the answer should be written in standard form.

Example: Add 3.2 x 10^4 and 6.5 x 10^3, thus:

$(3.2\times 10^{4})+ (6.5\times 10^{3})$

Make the powers of 10 equal. We can do this by changing 6.5 x 10^3 to 0.65 x 10^4.

Add the coefficients to get 3.2 + 0.65 = 3.85.

Represent the answer in standard form as 3.85 x 10^4, thus:

$3.85\times 10^{4}$

### 2. Multiplication:

• Multiply the coefficients.
• Add the powers of 10.
• The answer should be represented in standard form.

Example: Multiply 5.2 x 10^5 and 3.1 x 10^3

$\left ( 5.2\times 10^{5} \right )\times \left ( 3.1\times 10^{3} \right )$

Step 1. Multiply the coefficients to get 5.2 x 3.1 = 16.12.

Step 2. Add the powers of 10 (i.e., 5 + 3 = 8).

Step 2. Express the answer in standard form as 1.612 x 10^8, thus:

$1.612\times 10^{8}$

### 3. Division:

• Divide the coefficients.
• Subtract the denominator power of 10 from the numerator power of 10.
• The result should be written in standard form.

Example: Divide 6.4 x 10^6 by 2 x 10^3

$\left ( 6.4\times 10^{6} \right )\div \left ( 2\times 10^{3} \right )$

Divide the coefficients to get 6.4 ÷ 2 = 3.2.

Subtract the denominator power of 10 from the numerator exponent of 10 to get 10^(6-3) = 10^3.

Express the result in standard notation as 3.2 x 10^3, thus:

$3.2\times 10^{3}$

## Examples of Standard Form with their Solutions

1. Express 95300000 in standard form.

The standard form of the given number is 9.53 x 10^7, thus:

$9.53\times 10^{7}$

2. Write 0.0000000987 in standard form.

The standard form of the given decimal number is 9.87 x 10^8, thus:

$9.87\times 10^{8}$

3. Calculate (9.843 x 10^8) (5.4 x 10^7)

= (9.843 x 5.4) (10^8 + 7)

= 53.1522 x 10^15

= 5.31522 x 10^1 x 10^15

= 5.31522 x 10^16 or

$5.31522 \times 10^{16}$

4. Evaluate (7.43 x 10^-7) / (4.5 x 10^3)

= (7.43 ÷ 4.5) (10^-7 – 3)

= 1.651 x 10^-10 or

$1.651 \times 10^{-10}$

## Conclusion

As we learned in this article, the standard form is a way to represent both extremely large and small numbers. We detailed the method for changing whole and decimal numbers into standard form using clear steps, discussed arithmetic operations’ rules, and offered numerous examples for a better grasp of the concept.