# Scientific Notation

### Written by tutor Martina B.

Scientific notation is a convenient way of representing very small or very large values.

For example, mass of one electron is approximately:

0.0000000000000000000000000000009109 kg = 9.109x10^{-31} kg.

Mass of our beautiful planet Earth is approximately:

5,972,000,000,000,000,000,000,000 kg = 5.972x10^{24} kg.

Copying all these “leading” and “tailing” zeroes would prove impractical on calculators, in tabular representations, and especially on tests! Carrying out operations with such numbers would be very challenging. Imagine trying to compute the energy of a single photon of green light, with a wavelength of 555 nm:

E = h*c/

= (0.0000000000000000000000000000000006626 Js*300000000 m/s)/0.000000555 m

= 0.000000000000000000358 J

In scientific notation, the above expression becomes much simpler to evaluate:

E = h*c/

= (6.626*10^{-34} Js * 3.00*10^{8} m/s)/(5.55*10^{-7} m)

=3.58*10^{-19} J.

Any positive number N can be written in the following form:

**A * 10 ^{E}, where:
1 < A < 10 and
E is a positive or negative integer.
**

In the above notation, A represents the numerals of N with the decimal point placed __AFTER__ the first non-zero digit. For example, 7531 will have the decimal point between digits 7 and 5. Similarly, 0.0246 will have the decimal point between 2 and 4.

When a decimal point is placed behind the first non-zero digit, it is said to be in the standard position. Let N be any positive number, and let A be that number with the decimal point moved to the standard position. Then:

If 0 < N < 1, then N = A * 10^{negative exponent}

If 1 < N < 10, then N = A * 10^{0}

If N > 10, then N = A * 10^{positive exponent}

Moving the decimal point to its standard position will always produce a number between 1 and 10.

So, how do we convert 300,000,000 to scientific notation?

**First, starting at the decimal point of the given number, we count the number of decimal places we move, until we stop to the right of the first non-zero digit. The number of decimal places we move will be the exponent. If we move to the left, the exponent will be positive. If we move to the right, the exponent will be negative.
**

In our example, we will move the decimal 8 places to the left; therefore, the exponent is 8. The answer is:

3*10^{8}.

To convert 0.00357 to scientific notation, we will move the decimal 3 places to the right. The exponent is -3:

3.57*10^{-3}.

Remember to keep all significant figures when converting a number to scientific notation.
In the above example, 0.00357 has 3 significant figures, because the leading zeroes are place holders – not significant. Its scientific notation, 3.57*10^{-3}, also has three significant figures. Similarly, there is only one significant figure in 300,000,000, because the tailing zeroes are not significant. In scientific notation, it is perhaps more obvious that there is only one significant figure: 3*10^{8}.

The leading zeroes are never significant. The tailing zeroes are significant, if there is a decimal point placed anywhere in the number. For example:

200. = 2.00*10^{2}

70.00 = 7.000 * 10^{1}

In scientific notation, we keep the significant tailing zeroes. In fact, sometimes scientific notation is the only way to accurately represent the number of significant figures. Consider the following two numbers:

The number 300,000,000 with no decimal point has one significant figure.

The number 300,000,000. with a decimal point has nine significant figures.

A scientist may specify a level of precision, which is relevant for his measurement. He may specify the first three figures as significant, and the remaining figures are not significant.

How do we represent the number three hundred million with only three significant figures? The answer:

3.00*10^{8}.

Finally, examine the result of the following conversion of nm to m:

555 nm * (1*10^{-9}m/1nm) = 5.55*10^{-7}m.

Practice converting a number to scientific notation on your calculator. Learn to input scientific notation on your calculator. Simply read the instruction manual that came with your calculator or search online for your calculator model to find specific instructions.

### Scientific Notation Quiz

Convert the following numbers to scientific notation WITHOUT the calculator:

3.12

**A.**31.2*10

^{-1}

**B.**0.312*10

**C.**Already in scientific notation

**C**.

0.70

**A.**7.0*10

^{-1}

**B.**0.070*10

**C.**Already in scientific notation

**A**.

456789

**A.**45.6789*10

^{4}

**B.**4.56789*10

^{5}

**C.**Already in scientific notation

**B**.

0.027*10^{-9}

**A.**.27*10

^{-10}

**B.**2.7*10

^{-11}

**C.**Already in scientific notation

**B**.

531*10^{12}

**A.**53.1*10

^{13}

**B.**5.31*10

^{14}

**C.**Already in scientific notation

**B**.