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Scientific Notation

Scientific notation is the way that scientists easily handle very large numbers or very small numbers. For example, instead of writing 0.0000000056, we write 5.6 x 10-9. So, how does this work?

We can think of 5.6 x 10-9 as the product of two numbers: 5.6 (the digit term) and 10-9 (the exponential term).

Here are some examples of scientific notation.

10000 = 1 x 10424327 = 2.4327 x 104
1000 = 1 x 1037354 = 7.354 x 103
100 = 1 x 102482 = 4.82 x 102
10 = 1 x 10189 = 8.9 x 101 (not usually done)
1 = 100
1/10 = 0.1 = 1 x 10-10.32 = 3.2 x 10-1 (not usually done)
1/100 = 0.01 = 1 x 10-20.053 = 5.3 x 10-2
1/1000 = 0.001 = 1 x 10-30.0078 = 7.8 x 10-3
1/10000 = 0.0001 = 1 x 10-40.00044 = 4.4 x 10-4

As you can see, the exponent of 10 is the number of places the decimal point must be shifted to give the number in long form. A positive exponent shows that the decimal point is shifted that number of places to the right. A negative exponent shows that the decimal point is shifted that number of places to the left.

In scientific notation, the digit term indicates the number of significant figures in the number. The exponential term only places the decimal point. As an example,

46600000 = 4.66 x 107
This number only has 3 significant figures. The zeros are not significant; they are only holding a place. As another example,
0.00053 = 5.3 x 10-4
This number has 2 significant figures. The zeros are only place holders.

How to do calculations:

On your scientific calculator:

Make sure that the number in scientific notation is put into your calculator correctly.
Read the directions for your particular calculator. For inexpensive scientific calculators:

  1. Punch the number (the digit number) into your calculator.
  2. Push the EE or EXP button. Do NOT use the x (times) button!!
  3. Enter the exponent number. Use the +/- button to change its sign.
  4. Voila! Treat this number normally in all subsequent calculations.

To check yourself, multiply 6.0 x 105 times 4.0 x 103 on your calculator. Your answer should be 2.4 x 109.

On your cheap non-scientific calculator:

You will need to be familiar with exponents since your calculator cannot take care of them for you. For an introduction to rules concerning exponents, see the section on Manipulation of Exponents.

Addition and Subtraction:

Multiplication:

Division:

Powers of Exponentials:

Roots of Exponentials:

QUIZ:
Question 1Write in scientific notation: 0.000467 and 32000000
Question 2Express 5.43 x 10-3 as a number.
Question 3(4.5 x 10-14) x (5.2 x 103) = ?
Question 4(6.1 x 105)/(1.2 x 10-3) = ?
Question 5(3.74 x 10-3)4 = ?
Question 6The fifth root of 7.20 x 1022 = ?


Answers: (1) 4.67 x 10-4; 3.2 x 107 (2)0.00543 (3) 2.3 x 10-10 (2 significant figures) (4) 5.1 x 108 (2 significant figures) (5) 1.96 x 10-10 (3 significant figures) (6) 3.73 x 104 (3 significant figures)

Pick your next topic:
Algebraic Manipulation Significant Figures Manipulation of Exponents
Dimensional Analysis Logarithms The Quadratic Equation