Scientific notation

Scientific notation is a form of presenting very large numbers or very small numbers in a simpler form. As we know, the whole numbers can be extended till infinity, but we cannot write such huge numbers in a piece of paper. Also, the numbers which are present at the millions place after the decimal needed to be represented in a simpler form. Thus, it is difficult to represent a few numbers in their expanded form. Hence, we use scientific notations. Also learn, Numbers In General Form.

For example, 100000000 can be written as 108, which is the scientific notation. Here the exponent is positive. Similarly, 0.0000001 is a very small number which can be represented as 10-8, where the exponent is negative.

Scientific Notation Definition

As discussed in the introduction, the scientific notation helps us to represent the numbers which are very huge or very tiny in a form of multiplication of single-digit number and 10 raised to the power of the respective exponent. The exponent is positive if the number is very large and it is negative if the number is very small. Learn power and exponents for better understanding.

The general representation of scientific notation is:

a × 10b ; 1 ≤ a < 10

Scientific Notation Examples

The examples of scientific notation are:
490000000 = 4.9×108
1230000000 = 1.23×109
50500000 = 5.05 x 107
0.000000097 = 9.7 x 10-8
0.0000212 = 2.12 x 10-5

Also, read:

Scientific Notation Rules

To determine the power or exponent of 10, let us understand how many places we need to move the decimal point after the single-digit number.

  1. If the given number is multiples of 10 then the decimal point has to move to the left, and the power of 10 will be positive.
    Example: 6000 = 6 × 103 is in scientific notation.
  2. If the given number is smaller than 1, then the decimal point has to move to the right, so the power of 10 will be negative.
    Example: 0.006 = 6 × 0.001 = 6 × 10-3 is in scientific notation.

Positive and Negative Exponent

When the scientific notation of any large numbers is expressed, then we use positive exponents for base 10. For example:
20000 = 2 x 104, where 4 is the positive exponent.

When the scientific notation of any small numbers is expressed, then we use negative exponents for base 10. For example:
0.0002 = 2 x 10-4, where -4 is the negative exponent.

Problems and Solutions

Question 1: Convert 0.00000046 into scientific notation.

Solution: Move the decimal point to the right of 0.00000046 up to 7 places.

The decimal point was moved 7 places to the right to form the number 4.6

Since the numbers are less than 10 and decimal is moved to the right, so we use a negative exponent here.

⇒ 0.00000046 = 4.6 × 10-7

This is the scientific notation.

Question 2: Convert 301000000 in scientific notation.

Solution: Move the decimal to the left 8 places so it is positioned to the right of the leftmost non zero digits 3.01000000. Remove all the zeroes and multiply the number by 10.

Now the number has become = 3.01.

Since the number is greater than 10 and decimal is moved to left, therefore, we use here a positive exponent.

Hence, 3.01 × 108 is the scientific notation of the number.

Question 3:Convert 1.36 × 107 from scientific notation to standard notation.

Solution: Given, 1.36 × 107 in scientific notation.

Exponent = 7

Since the exponent is positive we need to move the decimal place 7 places to the right.

Therefore,

1.36 × 107 = 1.36 × 10000000 = 1,36,00,000.

Practice Questions

Problem 1: Convert the following numbers into scientific notation.

  1. 28100000
  2. 7890000000
  3. 0.00000542

Problem 2: Convert the following into standard form.

  1. 3.5 × 105
  2. 2.89 × 10-6
  3. 9.8 × 10-2