Scientific notation is a method of expressing numbers that are too big or too small to be conveniently written in decimal form.it is also referred to as ‘scientific form’ in Britain,  It is commonly used by scientists, mathematicians and engineers for complex calculations with lengthy numbers.On scientific calculators it is usually known as “SCI” display mode.

To write in scientific notation, follow the general form

 

 

where N is a number between 1 and 10, but not 10 itself, and m is any integer (positive or negative number).

 

In this article let us discuss what is scientific notation, what is the definition of scientific notation, scientific notation to standard form, scientific notation examples.

 

Scientific Notation Definition

Scientific notation is a method of expressing numbers in terms of a decimal number between 1 and 10, but not 10 itself multiplied by a power of 10.

The general for of scientific notation is

In scientific notation, all numbers are written in the general form as

N × 10m

N times ten raised to the power of m, where the exponent m is an integer, and the coefficient N is any real number. The integer m is called the order of magnitude and the real number N is called the significand. See the below figure –

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From the figure you can see that the exponent of 10 is the number of places the decimal point must be shifted to give the number in long form. A positive exponent shows that the decimal point is shifted to the right of that number and a negative exponent shows that the decimal point is shifted to the left of that number.

The digit term in the scientific notation indicates the number of significant figures in the number. The exponential term only places the decimal point. As an example, 

4660000 = 4.66 x 106

This number only has 3 significant figures. The zeros are not important, they are just placeholders. As another example,

0.00053 = 5.3 x 10-4

This number has 2 significant figures. The zeros are only place holders.

 

Scientific Notation Rules:

While writing the numbers in the scientific notation we have to follow certain rules they are as follows:

1. The scientific notations are written in two parts one is the just the digits, with the decimal point placed after the first digit, followed by multiplication with 10 to a power number of decimal point that puts the decimal point where it should be.

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2. If the given number is greater than 1 and multiples of 10 then the decimal point has to move to the left, and the power of 10 will be positive.
   Example: Scientific notation for 8000 will be 8 × 103.

3. If the given number is smaller than 1 means in the form of decimal numbers, then the decimal point has to move to the right, and the power of 10 will be negative.
   Example: Scientific notation for 0.008 will be 8 × 0.001 or 8 × 10-3.

 

Standard Form to Scientific Notation

To write 412,000,000,000 in scientific notation:

Use the general form N x 10m

Step1: Move the decimal place to the left to create a new number from 1 upto 10.
412,000,000,000 is a whole number, the decimal point will be given at the end of the number: 412,000,000,000.
So, you get N = 3.12.

 

Step2: Determine the exponent, it will be the number of times you moved the decimal.
Here, you moved the decimal 11 times and because you moved the decimal to the left, the exponent is positive. Therefore, m = 11, and so you get 1011

Step 3: Substitute the value of N and m in the general form of scientific notation

N x 10m

3.12 x 1011

Hence 3.12 x 1011 is in scientific form

Now write .00000041 in scientific notation.

Step 1: Move the decimal place to the right to create a new number from 1 upto 10.
So we get N = 4.1.

Step 2: Determine the exponent,it will be the number of times you moved the decimal.
Here, you moved the decimal 7 times and because you moved the decimal to the right, the exponent is negative. Therefore, m = –7, and so you get 10-7

 

Step 3: Substitute the value of N and m in the general form of scientific notation

N x 10m

4.1 x 10-7

Hence 4.1 is in scientific form.

 

Similarly scientific notations are converted to standard form.

Let us understand this with scientific notation examples. 

 

Solved Examples

1. Change scientific notation to standard form.

1.86 × 107

Solution:

 Given that 1.86 × 107 is in scientific notation.

Here Exponent m = 7

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

Therefore,

1.86 × 107 

= 1.86 × 10000000 

= 1,86,00,000.

 

2. Convert 0.0000078 into scientific notation.

Solution: 

Given that 0.0000078 is in standard form

To convert it in scientific notation use the general form

N x 10m

Move the decimal point to the right of 0.0000078 up to 6 places.

We get N = 7.8

Since the numbers are less than 10 we move the decimal point to the right, so we use a negative exponent here.

We get m = 10-6

therefore , 0.0000078 = 7.8 × 10-6

7.8 x 10-6 is the scientific notation.

 

Quiz Time:

1. Change scientific notation to standard form

1. 6.7 x 106

2. 4.5 x 10-9

2. Convert into scientific notations

1. 670000000000

2. 0.00000000089