Scientific Notation Rules and Examples

by
Ahmed Faizan

The goal of this module is to provide students with the tools they need to use scientific notation to represent quantities, apply electrical units of measure, convert metric units, and express measured data with the proper number of significant figures.

Objective

A learner will be able to:

  • Use scientific notation to represent quantities
  • Convert one electrical metric unit to another metric unit
  • Convert from one unit with a metric prefix to another unit using scientific notation
  • Express measurements with the proper number of significant figures.

Orienting Questions

  • How do you represent extremely large or small quantities using scientific notation?
  • What are the processes of performing arithmetic using scientific notation?
  • How do you convert measurements containing metric prefixes?

Introduction

When dealing with extremely large or small quantities, scientists and engineers use scientific notation as a form of representation.  In electronics, scientific notation is an important tool for representing electrical values. Important skills include being able to perform arithmetic (adding, subtracting, multiplying, and dividing) using scientific notation, and being able to convert between units of metric measures.

Quantities represented by the Scientific Notation

Very large and very small quantities are encountered a great deal in electronics.  Scientific notation is used in the place of huge numbers of digits. 

Scientific notation is a convenient way to express large or small numbers in order to perform arithmetic and other functions.  It uses a base number between 1 and 10, and a power of ten.  A power of ten is a representation of a base number of ten and an exponent indicating how many times the base number is raised. The power of ten is represented by a symbol written above and to the right of a digit, or an exponent.

For example, if we were to represent 230,000 in scientific notation, we would move the decimal point left until we have a number between 1 and 10 on the left side of the decimal.

In this case, we would move the decimal point between the 2 and 3.

Next, we would count the number of digits to the right of the decimal.  In our example, there are 5.  So 230,000 would be represented as 2.3 X 105.

Scientific notation can only have a number less than 10 to the left of the decimal.  Any numbers to the right of the decimal, greater than zero, must remain in the base number.  As in the above example, we left the 3 in the base number since it is greater than zero.

To convert a number represented by scientific notation to a decimal number, we would simply move the decimal to the right the number of places indicated by the exponent.

Example         

            Let’s take the following number and convert it to scientific notation:

2,500,000        Our number

2.5                   We place our decimal between the 2 and 5 which gives us our base number between 1 and 10.

2.5 X 106         We moved the decimal 6 places to the left.

 Small Numbers

When dealing with small numbers, the decimal is moved to the right.  Instead of a positive exponent (power of ten), it is negative.  This does not mean the number is negative.

For example, if we wanted to represent the quantity 0.00000362, we would move the decimal to the right until we have a number between 1 and 10.  In this case, our decimal would be between the 3 and 6.

Next, we would count how many digits are on the left side of the decimal.  In our example, we moved the decimal 6 places.  Our example would be 3.62 X 10-6.

Notice, we left the 2 because it is a number greater than zero.

To convert a small number represented by scientific notation to a decimal number, we move the decimal to the left the number of places indicated by the exponent.

Example

            Let’s represent the following decimal number in scientific notation:

0.000 000 025               our number.

2.5                               We moved the decimal point to the right to give us our base of 2.5 which is between 1 and 10.

2.5 X 10-8                      We moved the decimal 8 places to the right, giving us our exponent of (-8).

More examples

            516,570,000,000,000 = 5.1657 X 1014

            0.000100972 = 1.00972 X 10-4

            4,683.8 = 4.6838 X 103

            0.05871 = 5.871 X 10-2

           7.55 X 102 = 755                    

            190 X 106 = 190,000,000                   

            1.23 X 10-6 = 0.00000123                  

            9 X 10-3 = 0.009                     

View the video below before proceeding to the next section.

Scientific Notation Video

Arithmetic with scientific notation       

Scientific notation makes performing arithmetic simpler when dealing with very large and very small numbers.  This leaves less room for errors.

Addition

We add numbers in scientific notation using the following method:

  1. Express both numbers with the same power of ten.
  2. Add the base numbers.
  3. Bring the power of ten down to represent the new power of ten for the sum.
  4. Simplify so that the base number is between 1 and 10.

Example

How would we add 3 X 105 plus 6 X 104?

We need to first express the numbers using the same power of ten: 

            (3 X 105) + (60 X 105)

Add the base numbers:

            3 + 60 = 63

Bring the power of ten down:

            63 X 105

Simplify so that the base is a number between 1 and 10:

            6.3 X 106

Subtraction

The following method is used when subtracting powers of ten:

  1. Express both numbers with the same power of ten.
  2. Subtract the base numbers without their power of ten.
  3. Bring the power of ten down to represent the difference.
  4. Simplify so that the base number is between 1 and 10.

Example

Here is an example of subtracting numbers expressed in powers of ten:

            Subtract 3.5 X 10-12 from 9.5 X 10-11

First we represent both numbers in the same power of ten:

            (9.5 X 10-11) – (.35 X 10-11)

Subtract the base numbers:

            9.5 – .35 = 9.15

Bring down the power of ten:

             9.15 X 10-11 

Scientific Notation: Addition and Subtraction

Multiplication

To multiply numbers expressed in scientific notation use the following method:

  1. Multiply the base numbers without their powers of ten.
  2. Add the powers of ten using the algebraic rules for adding numbers (the exponents do not need to be the same).

Example

Multiply 6 X 103 by 4 X 10-5

Multiply the base numbers:        (6)(4) = 24

Add the exponents:                 3 + (-5) = -2

The product is:                        24 X 10-2

Simplified:                             2.4 X 10-1

Division

To divide numbers expressed in scientific notation use the following method:

  1. Write the problem as a fraction with a numerator and denominator.
  2. Divide the base numbers.
  3. Subtract the exponent in the denominator from the exponent in the numerator (the exponents do not need to be the same).

Example

Divide 7 X109 by 3.5 X 104:

            Represent problem as a fraction:      

$\frac{7~X~{{10}^{9}}}{3.5~X~{{10}^{4}}}$

            Divide the base numbers:

                                                                        7 / 3.5 = 2

            Subtract the exponents:

                                                                        9 – 4 = 5

The quotient is:                                  2 X 105

Scientific Notation: Multiplication and Division

Converting Measures with Metric Prefixes

In the electronics field, you will deal with measureable quantities.  You will measure voltage, current, and resistance as well as many other electrical quantities.  All of these measurements have certain units and symbols that are used in combination with engineering notation.

Engineering Notation

Similar to scientific notation, engineering notation uses the same “power of ten” concept.  A difference is that engineering notation can have up to three digits to the left of the decimal.  Also, engineering notation can only have exponents that are multiples of three (3, 6, 9, etc.).

Example

Below are a few examples of numbers represented by both scientific and engineering notations:

Number                                   Scientific notation                  engineering notation 

23,000                                     2.3 X 104                                       23 X 103

500                                          5 X 102                                    500 or .5 X 103

0.000052                                 5.2 X 10-5                                52 X 10-6

Electrical Units

Electrical units and quantities are represented by a letter symbol.  Below is a table of some common electrical quantities, SI (International Standard), and symbols:

QUANTITY SYMBOL SI UNIT SYMBOL
Voltage V Volt V
Current I Ampere (Amp) A
Charge Q Coulomb C
Resistance R Ohm Ω
Capacitance C Farad F
Inductance L Henry H
Power P Watt W
Energy W Joule J
Time T Seconds S
Frequency F Hertz Hz 


Primer on Electrical Units, Abbreviations and Symbols 1-2

Metric Prefixes

Metric prefixes represent some of the most common powers of ten in engineering notation.  Below is a table showing the most common metric prefixes:

Prefix Prefix
Symbol		 Value
Pico P 10-12 = 0.000 000 000 001
nano n 10-9 = 0.000 000 001
micro µ 10-6 = 0.000 001
milli m 10-3 = 0.001
kilo k 103 = 1000
Mega M 106 = 1000 000
Giga G 109 = 1000 000 000
Tera T 1012 = 1000 000 000 000

Example

            Show the following number with the prefix and the unit symbols:

                        0.005 Volts     Our number

                        5 X 10-3           Volts in scientific notation

                        5 m Volts        On our table we see 10-3 is represented by m

                        5 mV               The symbol for volts is V

Converting Metric Units

In order to do some calculations involving metric units, it is more convenient to convert the metric prefixes.  There are some basic guidelines to follow when converting prefixes:

  1. Move the decimal point to the right when converting large units to smaller units.
  2. Move the decimal point to the left when converting small units to larger units.
  3. Find the difference of the powers of ten to decide how many places to move the decimal point.

Example

  1. Convert 3 millifarads to microfarads.

Using the table above, we see that mF is millifarads (10-3).  Microfarads is 10-6.  Since microfarads are smaller than millifarads we would move the decimal three places to the right.  This would give 300µF.

  1. Convert 4000 nanoamperes to microamperes.

                   We would move the decimal three places to the left.

                  4000nA = 4000 X 10-9A = 4 X 10-6 = 4µA

  1. Convert 1600 Kilohms to Megaohms

We would move the decimal three places to the left.

1600kΩ = 1600 X 103 = 1.6 X 106  = 1.6MΩ

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