# Scientific Notation

Scientific notation is a special way of representing numbers which are too large or small in a unique way that makes it easier to remember and compare them. They are expressed in the form (a Ã— 10n). Here â€˜aâ€™ is the coefficient, and â€˜nâ€™ is the power or exponent of the base 10.

The diagram below shows the standard form of writing numbers in scientific notation:

Thus, scientific notation is a floating-point system where numbers are expressed as products consisting of numbers between 1 and 10 multiplied with appropriate power of 10. It helps to represent big and small numbers in a much easier way.

The speed of light(c) measured in a vacuum is approximately 300,000,000 meters per second, which is written as 3 Ã— 108 m/s in scientific notation. Again the mass of the sun is written as 1.988 Ã— 1030 kg. All these values, if written in scientific form, will reduce a lot of space and decrease the chances of errors.

## Scientific Notation Rules

We must follow the five rules when writing numbers in scientific notation:

1. The base should always be 10
2. The exponent (n) must be a non-zero integer, positive or negative
3. The absolute value of the coefficient (a) is greater than or equal to 1, but it should be less than 10 (1 â‰¤ a < 10)
4. The coefficient (a) can be positive or negative numbers, including whole numbers and decimal numbers
5. The mantissa contains the remaining significant digits of the number

## How to Do Scientific Notation with Examples

As we know, in scientific notation, there are two parts:

• Part 1: Consisting of just the digits with the decimal point placed after the first digit
• Part 2: This part follows the first part by Ã— 10 to a power that puts the decimal point where it should be

While writing numbers in scientific notation, we need to figure out how many places we should move the decimal point. The exponent of 10 determines the number of places the decimal point gets shifted to represent the number in long form.

There are two possibilities:

### Case 1: With Positive Exponent

When the non-zero digit is followed by a decimal point

For example, if we want to represent 4237.8 in scientific notation, it will be:

• The first part will be 4.2378 (only the digit and the decimal point placed after the first digit)
• The second part following the first part will be Ã— 103 (multiplied by 10 having a power of 3)

### Case 2: With Negative Exponent

When the decimal point comes first, and the non-zero digit comes next

For example, if we want to represent 0.000082 in scientific notation, it will be:

• The first part will be 8.2 (only the coefficient in decimal form and the decimal point placed after the first digit)
• The second part following the first part will be Ã— 10-5 (multiplied by 10 having a power of -5)

Here is a table showing some more examples of numbers written in scientific notation:

Let us solve some more word problems involving writing numbers in scientific notation.

Write the number 0.0065 in scientific notation.

Solution:

0.0065 is written in scientific notation as:
6.5 Ã— 10-3

Convert 4.5 in scientific notation.

Solution:

4.5 is written in scientific notation as:
4.5 Ã— 100

Write 53010000 in scientific notation.

Solution:

53010000 is written in scientific notation as:
5.301 Ã—107

Light travels with a speed of 1.86 x 105 miles/second. It takes sunlight 4.8 x 103 seconds to reach Saturn. Find the approximate distance between Sun and Saturn. Express your answer in scientific notation.

Solution:

As we know,
Distance (d) = Speed (s) Ã— Time (t), here s = 1.86 x 105 miles/second, t = 4.8 x 103 seconds
= 8.928 x 108 miles

## Other Ways of Writing in Scientific Notation

We sometimes use the ^ symbol instead of power while writing numbers in scientific notation. In such cases, the above number 4237.8, written in scientific notation as 4.2378 Ã— 103, can also be written as 4.2378 Ã— 10^3 Similarly, calculators use the notation 4.2378E; here, E signifies 10 Ã— 10 Ã— 10

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