The easiest way to write the very large and very small numbers is possible due to the scientific notation. For example, the number 2500000000000000000000 is too large and writing it multiple times requires a short-hand notation called *scientific notation*.

The scientific notation involves the smallest number as possible (between 1 and 10) multiplied by (using the '\(\times \)' sign) the power of 10. The above number is represented in scientific notation as \(2.5\times {{10}^{21}}\). You can also write the number as \(250\times {{10}^{19}}\) but it's going to remove its name, the short-hand notation!

Generally you use the smallest number as 2.5 which is then multiplied by the appropriate power of 10. When you do the real multiplication between the smallest number and the power of 10, you obtain your number.

Another example is for small numbers. Consider 0.00000000000000000000453 and this can be written in the scientific notation as \(4.53\times {{10}^{-23}}\). Scientific notations are frequently used in calculations with large or small numbers in physics.

## What are scientific notation rules?

The scientific notation is expressed in the form \(a \times 10^n\) where \(a\) is the coefficient and \(n\) in \(\times 10^n\) (power of 10) is the exponent. The coefficient is the number between 1 and 10, that is \(1 < a < 10\) and you can also include 1 (\(1 \geq a < 10\)) but 1 is not generally used (instead of writing 1, it's easier to write in power of 10 notation). The scientific notation is the way to write very large and very small numbers in practice and it is applied to positive numbers only. The rules to convert a number into scientific notation are:

- First thing is we determine the coefficient. Convert the number into greater than 1 and smaller than 10 by placing the decimal point at appropriate location (only one nonzero number exists to the left of the decimal point), and remove any trailing or leading zeros. For example, if 3453000 is the number, convert it to 3.453.
- Move either to the right or to the left (depending on the number) across each digit to the new decimal location and the the number places moved will be the exponent. In 3453000, we move from the right end and number of places we move to our new location is 6, so 6 will be the exponent.
- If the original number is less than 1 (x < 1), the exponent is negative and if it is greater than or equal to 10 (x \(\geq\) 10), the exponent is positive. If it is between 1 and 10 including 1 (1 \(\geq\) x < 10), the exponent is zero. In 3453000, the exponent is positive.
- Alternatively you can say the rule number 3 as, if you move to the right, the exponent is negative and if you move to the left, the exponent is positive. And if you do not move at all, the exponent is zero but you do not need to express such number in scientific notation. Note that the scientific notation is the way to express very small and very large numbers easily.
- Add the \(\times 10^n\) part.

The above rules are more elaborated in the examples given below.

### Example 1

Let's turn this number 0.00000026365 into scientific notation. First convert this number to greater than 1 and smaller than 10. To do that you you just need to add a decimal point between 2 and 6. The new number is 2.6365. The final step is to count the number of steps (places) we need to move to the right from the old decimal location to the new location as shown in Figure below.

The exponent is the negative of the number of steps (number of places) we moved to the right of decimal point to our new location. Hence the number in scientific notation is \(2.6365 \times 10^{-7}\). The figure above explains this more clearly.

### Example 2

Now you have a large number 3424300000 and you want to express this number in scientific notation. To convert this number to a number smaller than 10 and greater than 1 you just need to add decimal point between 3 and 4 and the number without leading zeroes becomes 3.4243.

Now you got the new location of decimal point. Simply move to the left from the right end of the number to the new decimal location. Note that this is a whole number and the decimal point is understood to be at the right end (3424300000.). Count the number of digits you moved across and that number will be exponent. So the number in scientific notation is \(3.4243 \times 10^{9}\).

## How to remove scientific notation?

Now we convert numbers already in scientific notation to their original form. You can follow some easy steps to successfully convert the number in scientific notation back to normal form. Here are the rules.

- If the exponent is positive, move to the right the number of decimal places expressed in the exponent.
- If the exponent is negative, move to the left the number of decimal places expressed in the exponent.
- If there are not enough digits to move across, add zeros in the empty spaces.
- You do not need the \(\times\) 10 anymore and remove it.

This is quiet easy. The exponent tells you the number of decimal places to move. All you have to do is move either to the right or to the left across digits. If there is no digit to move across, add zero in the empty place until you complete.

### Example 1

We consider a number 3.456 \(\times\) 10\(^7\) and convert it to original number without scientific notation. The exponent is 7 so we move 7 steps to the right of the current decimal location. The figure shows you the way to move.

After moving across three digits, there are no more digits to move but we add 0 in empty places and you get the original number, 34560000.

### Example 2

Let's consider a small number with negative exponent, \(7.312 \times 10^{-5}\). Now you move to the left of decimal location 7 times. Here moving means we are taking the decimal point to the new location. So the number without scientific notation is .00007312 or 0.00007312 (the zero before the decimal point is optional).