Scientific Notation

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.

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:

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. Add the $\times 10^n$ part.

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

Scientific Notation Example 1

You have a number 0.00000026365 and you want to write this number in 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.

Figure 1 Conversion to scientific notation.

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.

Scientific Notation 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.

Figure 2 Conversion to scientific notation.

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 $3.4243 \times 10^{9}$.

Removing 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.

  1. If the exponent is positive, move to the right the number of decimal places expressed in the exponent.
  2. If the exponent is negative, move to the left the number of decimal places expressed in the exponent.
  3. If there are not enough digits to move across, add zeros in the empty spaces.
  4. 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.

Removing Scientific Notation 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.

Figure 3 Removing scientific notation.

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

Removing Scientific Notation 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).

Figure 4 Removing scientific notation

Let's look at the addition, subtraction, multiplication and division of numbers in scientific notation. The arithmetic with numbers in scientific notation is similar to the arithmetic of numbers without scientific notation. One difference is that the rules of exponent applies with scientific notation.

Addition in Scientific Notation

The addition in scientific notation can be done by following very simple rules:

  1. Change all numbers to the same power of 10.
  2. Add the coefficients and put the common power of 10 as $\times 10^n$.
  3. Convert to scientific notation again if there is not only one nonzero number to the left of decimal point. If you need to do this, change or add the exponents again (apply exponents rule).

Scientific Notation Addition Example

You have two numbers $2.4 \times 10^3$ and $5.71 \times 10^5$. To add these two numbers easily, you need to change all numbers to the common power of 10. You can change exponent of any number. Here we change the exponent in $5.71 \times 10^5$ to 3 and it is $571 \times 10^3$ (note the decimal point moved two places to the right).

\[\begin{align*} 2.4 \times 10^3 + 5.71 \times 10^5 \\ 2.4 \times 10^3 + 571 \times 10^3 \\ (2.4 + 571) \times 10^3 \\ 573.4 \times 10^3 \\ 5.734 \times 10^2 \times 10^3\\ 5.734 \times 10^{2+3} \\ 5.734 \times 10^5 \end{align*}\]

Now simply add coefficients, that is 2.4 + 571 and put the power 10, so the number after addition is $573.4 \times 10^3$. The final step is to convert this number to the scientific notation. Note that the coefficient must be greater than 1 and smaller than 10 in scientific notation. So the number in scientific notation after the addition is $5.734 \times 10^5$.

You do not need to convert the final number into scientific notation again if you have changed exponent in $2.4 \times 10^3$ to 5, so it is a good idea to convert smaller exponent to greater exponent. After subtracting the two exponents 5 - 3 you get 2 and the 2 to the power of 10 is 100. So 2.4 needs to be divided by 100 or the decimal point needs to be moved two places to the left, and that gives 0.024. Now we have the same exponent in both numbers. Just add 0.024 + 5.71 which gives 5.734 and the result is $5.734 \times 10^5$.

\[\begin{align*} 2.4 \times 10^3 + 5.71 \times 10^5 \\ 0.024 \times 10^3 + 5.71 \times 10^5 \\ (0.024 + 5.71) \times 10^5 \\ 5.734 \times 10^5 \\ \end{align*}\]

Multiplication in Scientific Notation

Multiplication of numbers in scientific notation is easy. Simply multiply the coefficients and add the exponents. If necessary, change the coefficient to number greater than 1 and smaller than 10 again.

Multiplication in Scientific Notation Example

Here we have two numbers $7.23 \times 10^{34}$ and $1.31 \times 10^{11}$. When you multiply these two numbers, you multiply the coefficients, that is $7.23 \times 1.31 = 9.4713$. Then all exponents are added, so the exponent on the result of multiplication is $11+34 = 45$. The final result after the multiplication is $9.4713 \times 10^{45}$ or the process is shown below:

\[(7.23 \times 10^{34}) \times (1.31 \times 10^{11}) \\ 7.23 \times 1.31 \times 10^{34} \times 10^{11} \\ 9.4713 \times 10^{34 + 11}\\ 9.4713 \times 10^{45}\]

Apply the exponents rule and voila! If the coefficient in the result is greater than 10 convert that number to greater than 1 and smaller than 10 by changing the decimal location and add the exponents again.

Division in Scientific Notation

The division of two scientific numbers is similar to multiplication but in this case we divide coefficients and subtract the exponents.

Division in Scientific Notation Example

You have two numbers $1.03075 \times 10^{17}$ and $2.5 \times 10^5$ . To divide these numbers we divide 1.03075 by 2.5 first, that is 1.03075/2.5 = 0.4123. Then we subtract the exponents of these numbers, that is 17 - 5 = 12 and the exponent on the result of division is 12.

Note that the number 0.4123 is less than 1, so we make this number greater than 1 and smaller than 10. To do that the decimal point goes between 4 and 1 and the number of steps we moved to the right across the digits to our new location is subtracted from the exponent of 10. So, The final exponent of 10 is $12 - 1 = 11$ and the number is 4.123. So the result is $4.123 \times 10^{11}$. Or mathematically,

\[\begin{align*} \frac{1.03075 \times 10^{17}}{2.5 \times 10^5} &= \frac{1.03075}{2.5} \times 10^{17 - 5} \\ &= 0.4123 \times 10^{12} = 4.123 \times 10^{-1} \times 10^{12} \\ &= 4.123 \times 10^{-1+12} = 4.123 \times 10^{11} \end{align*}\]

Significant Figures in Scientific Notation

How to determine the significant figures of very large and very small numbers? One benefit of scientific notation is you can easily express the number in the correct number significant figures.

For example, you are not sure that this number 17100000000000 has two, three or five significant figures. If this number has two significant figures, this number can be expressed in scientific notation as $1.7 \times 10^{13}$. If this number has five significant figures, it can be expressed in scientific notation as $1.7100 \times 10^{13}$.

How to Do Scientific Notation Using Calculators?

All scientific calculators allow you to express numbers in scientific notation and do calculation. The buttons to express numbers in scientific notation in calculators look like EXP, EE, $\times 10^{n}$ etc. The button depends on the make and model of your calculator but the function is the same in all calculators.

For example, in some calculators if you want to write $1.71 \times 10^{13}$ in scientific notation you write 1.71E13 using the button EXP or EE in the display screen. The button EXP or EE display E or e in calculator screen which represents the exponent. Other buttons such as $\times 10^n $ or $\times 10^x$ etc allow you to add exponent directly in the exponent form including the $\times 10$. Explore a little bit in your calculator and you'll be easily able to do calculations involving scientific notation.

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