Scientific Notation Explained with Rules and Applications

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 the scientific notation, the definition of scientific notation, a scientific notation to standard form, and 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 × 10^m

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.

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 placeholders.

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.

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 10^m

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 = 4.12.

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 10^m

4.12 x 10¹¹

Hence 4.12 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 10^m

4.1 x 10^-7

Hence 4.1 x 10-7 is in scientific form.

Similarly, scientific notations can be converted to standard form.

Let us understand this with help of examples. 

Scientific Notation to Standard Form 

To write 5.56 × 104 in standard form

Given that 5.56 × 104 is in scientific notation.

Here Exponent m = 4

Since the exponent is positive we need to move the decimal point to 4 places to the right.

Therefore,

5.56 × 104 

= 5.56 × 10000 

= 55,600.

So, the standard form is 55,600.

Solved Examples

1. Change scientific notation to standard form of 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.

So, the standard form is 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 10^m

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 1 we move the decimal point to the right, So we use a negative exponent here.

We get m = -6

Put the value of N and m in general form

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

Importance of Scientific Notation

Scientific Notation is a manner in which all scientists easily handle very large numbers or the very small numbers.  Any number can be written in scientific notation when it falls between 1 and 10 and is multiplied by a power of 10.  It is used globally by engineers, mathematicians and statisticians for important calculations and denotations. It is of great significance for the purpose of representing numbers.

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