Scientific Notation

Some physical quantities are too small and some are extremely large and expressing these magnitudes would be too long. A way of expressing these magnitudes efficiently is through scientific notation. In this way, a magnitude of certain quantity will be easier to read, easier to write and easier to understand compared to very long numbers.

To express numbers in scientific notation, the following rules are to be followed:

  1. Show the value with the correct number of significant figures.
  2. For very large numbers, or the whole numbers, move the decimal point to the left and stop before the last nonzero digit of the number. For very small numbers, or the decimal numbers, move the point to the right and stop right after the first nonzero digit.
  3. Show the number being multiplied by ten raised to the correct power. The power corresponds to the number of moves of the decimal point.

Example 1:

The speed of light is said to be approximately equal to 300, 000, 000 m/s. How is this value expressed in scientific notation?

Answer:

Since the given is a whole number, the decimal point is moved to the left and stop right before 3 which is the last nonzero digit. Thus, the scientific notation for the speed of light can be expressed as 3 x 108 m/s.


Example 2.

The charge of an electron is approximately equal to -0.0000000000000000001602 C. Express this value in scientific notation.

Answer: Since the value is a decimal number, the decimal point will be moved to the right and stop right after 1. The number of moves of the decimal point is 19, thus, the charge of the electron in scientific notation is written as \(-1.602 \times 10^{-19}\) C.


Example 3.

The Earth's diameter is said to be equal to 12,742,000 in meters. How is this value written in scientific notation?

Answer: By moving the decimal point to the left, we obtain the answer as \(1.2742 \times 10^7\;m\).


Example 4.

The universal gravitational constant, G, is equal to \(6.67 \times 10^{-11}\;Nm^2/kg^2\). Express this value as simple number.

Answer: Since the power is -11, it indicates that the decimal point is moved 11 places to the right. To write it back as simple number, we will add zeros to the left of the first nonzero digit then move the decimal point 11 places to the left. Thus, the value of the universal gravitational constant is \(0.0000000000667\;m.\)


Example 5.

A 40' Dry Cargo Hi-Cube container has a cargo capacity of \(2.6330 \times 10^4\;kg\). How is this value written as simple number?

Answer: Note that the power is 4, thus the given value is a large number. To write the value in simple number, move the decimal place 4 places to the right up to the last digit. Therefore, the cargo capacity of the container is 26, 330 kg.


Operations with Scientific Notation

Simple arithmetic does not apply directly in scientific notation. There are different ways and rules in adding, subtracting, multipying and dividing scientific notation.


Addition and Subtraction

To add or subtract scientific notation,

  1. Convert all the numbers to the same power of 10.
  2. Add or subtract only the digit term.
  3. Copy the exponential term.

Example 6.

\(1.23 \times 10^{-3} + 4.5 \times 10^{-5}=1.23 \times 10^{-3} +0.045 \times 10^{-3}=1.275\times 10^{-3}\)


Multiplication

To multiply scientific notation,

  1. Multiply the digit term.
  2. Add the exponents of the exponent term.
  3. In case of more that one nonzero digit to the left of the final answer, the decimal point is moved so that there is only one nonzero digit to the left of the decimal point. The number of moves will be added to the exponent.

Example 7.

Multiply \(6.7 \times 10^8\) and \(8.91 \times 10^{10}\).

Solution:

\((6.7 \times 10^8)(8.91 \times 10^{10})=59.697\times10^{18}=5.9697\times10^{19}\)


Division

In dividing scientific notation,

  1. Divide the digit term.
  2. Subtract the exponents of the exponent term.
  3. In case of more that one nonzero digit to the left of the final answer, the decimal point is moved so that there is only one nonzero digit to the left of the decimal point. The number of moves will be added to the exponent.

Example 8.

\(9.75 \times 10^{-9}/3.20\times10^{-3}=3.046875\times10^{-6}\)