Scientific notation

Scientific notation is a way of representing very large or very small numbers. The given number is written as the product of a decimal number, between 1 and 10, which is called a coefficient, and a power of 10. It has been one of the ancient mathematical methods. If numbers are too big or too small to be calculated, by using scientific representations of numbers, we can solve the mathematical operation easily. Here we will learn about the conversion of standard or decimal notation to scientific one along with its rules and examples.

Scientific Notation Conversion

Scientific notation is a way to express very large or very small numbers to perform the calculation. It is based on the powers or exponents of the base number 10. It is just a way of expressing huge numbers like 1,000,000 or uncommonly small numbers like 0.000000001. It converts a decimal number into a product of a number between 1 and 10, and a power of 10.

Generally, the Scientific Notation of a number is given by;

a × 10b ; 1 ≤ a < 10

‘a’ is the digit, called mantissa, which ranges from 1 to less than 10 and ‘b’ is the power or exponent, which shows exactly how many places to move the decimal point.

Scientific Notation Rules

Now, to figure out the power of 10, let’s see how many places do we move the decimal point.

  1. If the given standard number is 10 or greater than 10, then the decimal point has to move to the left, and the power of 10 will be positive.

Example: 4000 = 4 × 103 is in scientific notation.

  1. If the given standard number is smaller than 1, then the decimal point has to move to the right, so the power of 10 will be negative.

Example: 0.004 = 5 × 0.001 = 4 × 10-3 is in scientific notation.

Scientific Notation Examples

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