Scientific notation is a standard way of writing very large and very small numbers so that they’re easier to both compare and use in computations.
Every number in the scientific notation must be in the form of
a x 10^{n}
where 1 ≤ a < 10 and n must be a positive or negative integer.
To convert a number to scientific notation, first we have to identify where the decimal point and non zero digit come.
There are two cases in it.
Case 1 :
To move the decimal point to the left, we have to count number of digits as explained in the example given below.
According to the example given above, we have to move the decimal point 3 digits to the left and exponent of 10 should be 3 (positive integer)
When we do so, we get the scientific notation of the given number.
So,
2301.8 = 2.3018 x 10^{3}
Case 2 :
To move the decimal point to the right, we have to count number of digits as explained in the example given below.
According to the example given above, we have to move the decimal point 5 digits to the right and exponent of 10 should be -5 (negative integer)
When we do so, we get the scientific notation of the given number.
So,
0.000023 = 2.3 x 10^{-5}
Important Note:
If we don't find decimal point at anywhere of the given number, we have to assume that there is decimal point at the end of the number.
For example,
2300000 -------------> 2300000.
Here, the non zero digit comes first and decimal point comes next. So we have to apply case 1 to convert this number into scientific notation.
To convert a number from scientific notation to standard notation, first we have to notice the exponent of 10 in scientific notation.
If the exponent of 10 is positive, we have to move the decimal point to the right.
For example, if you have 10^{3}, you have to move the decimal point 3 digits to the right.
If the exponent of 10 is negative, we have to move the decimal point to the right.
For example, if you have 10^{-5}, you have to move the decimal point 5 digits to the left.
Examples 1-3 : Write the given number in standard form.
Example 1 :
5.236 x 10^{5}
Solution :
Here, the exponent of 10 is positive 5. So we have to move the decimal point five digits to the right.
In 5.236, we have only three digits after the decimal point.
So, we have to add two zeros to move the decimal point five digits to the right.
Therefore, the standard form of 5.236 x 10^{5} is
523600
Example 2 :
6.415 x 10^{-6}
Solution :
Here, the exponent of 10 is negative 6. So we have to move the decimal point six digits to the left.
In 6.415, we have only 1 digit before the decimal point.
So, we have to add five zeros to move the decimal point six digits to the left.
Therefore, the standard form of 6.415 x 10^{-6} is
0.000006415
Example 3 :
2.0 x 10^{7}
Solution :
Here, the exponent of 10 is positive 7. So we have to move the decimal point seven digits to the right.
In 2.0, we have one zero to the right of the decimal point.
So, we have to add six more zeros to move the decimal point seven digits to the right.
Therefore, the standard form of 2.0 x 10^{7} is
20000000
Examples 4-6 : Write the given number in scientific notation.
Example 4 :
4580245
Solution :
Here, we don't find decimal point in 4580245. So we have to assume that there is decimal point at the end .
Then,
4580245 ---------> 4580245.
Here, non zero digit comes first and decimal point comes next.
We have to move the decimal point to the left.
There are six digits between the 1^{st} non zero digit and the decimal point.
So, the decimal point has to be moved six digits to the left and exponent of 10 should be 6 (positive integer).
Therefore, the scientific notation of 4580245 is
4.4580245 x 10^{6}
Example 5 :
325.8645
Solution :
325.8645
Here, non zero digit comes first and decimal point comes next.
We have to move the decimal point to the left.
There are two digits between the 1^{st} non zero digit and the decimal point.
So, the decimal point has to be moved two digits to the left and exponent of 10 should be 2 (positive integer).
Therefore, the scientific notation of 325.8645 is
3.258645 x 10^{2}
Example 6 :
0.0008743
Solution :
Here, decimal point comes first and non zero digit comes next.
We have to move the decimal point to the right.
There are for digits from the decimal point up to the first 1^{st }non zero digit.
So, the decimal point has to be moved four digits to the right and exponent of 10 should be -4 (negative integer).
Therefore, the scientific notation of 0.0008743 is
8.743 x 10^{-4}
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