6. Scientific Notation
Scientific notation (also known as standard form) is used for numbers that are either very large or very small, like the following.
The distance from Earth to the nearest star system, Alpha Centauri, is about
39 900 000 000 000 000 m
Earth's mass is about:
6 000 000 000 000 000 000 000 000 kg
Red light has a wavelength of about
0.000 000 7 m
Scientific notation is a more convenient way of expressing such large or small numbers. It is used extensively in science and engineering, and makes life easier when working with large (or small) numbers on calculators or computers.
Scientific Notation - Definition
A number in scientific notation is expressed as
P × 10k
where: 1 ≤ P < 10 and k is an integer
Example 1
(a) 340 000 000 (ordinary notation)
= 3.4 × 100 000 000
= 3.4 × 108 (scientific notation)
(b) 0.000 015 (ordinary notation)
= 1.5 ÷ 100 000
= 1.5 × 10-5 (scientific notation)
(c) 68 250 000 (ordinary notation)
= 6.825 × 10 000 000
= 6.825 × 107 (scientific notation)
To change a number from scientific notation to ordinary notation, we simply reverse the procedure shown in the above examples.
Example 2
Using scientific notation, we can write our earlier examples as:
Distance to Alpha Centauri:
39 900 000 000 000 000 m = 3.99 × 1016 mEarth's mass:
6 000 000 000 000 000 000 000 000 kg = 6 × 1024 kgWavelength of red light:
0.000 000 7 m = 7 × 10−7 m
Example 3
We are all affected by slow Internet connections, but German scientists in 2011 succeeded in transmitting 26 terabits per second on a single laser beam. A "bit" is the smallest unit of information (a "0" or a "1"). That number is:
26 terabits = 26 000 000 000 000 = 2.6 × 1013 bits
This is the equivalent of being able to transmit 700 DVDs per second!
Example 4
(7 × 1011) × (6 × 10-3)
= 42 × 108 (This is not in scientific notation!)
= 4.2 × 109
Exercises
Perform the following calculations using a calculator, by first expressing all numbers in scientific notation:
(1) 0.000 058 × 4 000 000 000 000
Answer
0.000 058 × 4 000 000 000 000
= (5.8 × 10−5) × (4 × 1012)
= 2.32 × 108
[Considering the number of significant digits in the question, strictly our answer should be: 2 × 108,]
(2) 35 000 000 000 ÷ 0.000 000 000 000 07
Answer
35 000 000 000 ÷ 0.000 000 000 000 07
= (3.5 × 1010) ÷ (7 × 10−14)
= 5 × 1023
[Note that a large number divided by a small number gives a very large number!]
(3) 0.000 000 000 000 06 ÷ 53 000 000 000 000
Answer
0.000 000 000 000 06 ÷ 53 000 000 000 000
= (6 × 10−14) ÷ (5.3 × 1013)
= 1.13207547 × 10−27
[Considering the number of significant digits in the question, strictly our answer should be: 1 × 10−23. Note also that a small number divided by a big number is a very small number.]
Scientific Notation in the binary number system
We can extend the concept of scientific notation to other number systems.
See how this works in IntMath Newsletter: Binary scientific notation.