When we are discussing electromagnetic radiation we are often dealing with both very large and very small numbers. Therefore scientific notation is often used to write these numbers.
Many of the variables for radiation equations are written in scientific notation, so you will be to become comfortable converting numbers to and from scientific notation as well as adding, multiplying and dividing numbers in scientific notation.

Converting from "Normal" to Scientific Notation

In scientific notation a positive exponent indicates that the decimal point is moved that number of places to the right. A negative exponent indicates that the decimal point is shifted that number of places to the left.
When converting to scientific notation place the decimal point after the first non-zero digit, and count the number of places the decimal point has moved.
If the decimal place has moved to the left then multiply by a positive power of 10; to the right will result in a negative power of 10.

Example: Write 3040 in scientific notation Move the decimal point 3 places to the left, so it becomes 3.04 x 10^{3}

Example: Write 0.00012 in scientific notation Move the decimal point 4 places to the right, so it becomes 1.2 x 10^{-4}

Converting from Scientific Notation to "Normal"

If the power of 10 is positive, then move the decimal point to the right; if it is negative, then move it to the left.

Example: Convert 4.01 x 10^{5} to "normal notation" Since the exponent is positive we move the decimal point five places to the right, so it becomes 401,000

Working with Exponent & Numbers in Scientific Notation

Exponents and numbers in scientific notation are frequently used in remote sensing applications. Whether solving for the frequency of electromagnetic radiation, or using Wien's Law to determining the radiation of maximum radiation, you will find multiple variables with exponents. Therefore it's important that to understand the rules for working with exponents, this will also make solving the equations much easier.

Multiplication

To multiply two numbers expressed in scientific notation, simply multiply the coefficient (numbers in front) and then add the exponents.

Example: Calculate (5.1 x 10^{4}) • (2.5 x 10^{3}) Multiply the coefficients, then add the exponents. In this case we end up with more that one digit in front of the decimal. Therefore we need to move the decimal to the left one place, which adds one to the exponent

Division

To divide two numbers expressed in scientific notation, divide the coefficient (numbers out front) and subtract the exponents.

Example: Calculate (6.2 x 10^{6}) / (3.1 x 10^{3}) Divide the coefficients, then subtract the exponents.

There are also exponent rules that apply to powers and negative exponents. The table below is a quick reference guide to all of the commonly exponent rules and properties

Table of Exponents rules and properties

Rule name

Rule

Example

Product rules

a^{ n} ⋅ a^{ m} = a^{ n+m}

2^{3} ⋅ 2^{4} = 2^{3+4} = 128

a^{ n} ⋅ b^{ n} = (a ⋅ b)^{ n}

3^{2} ⋅ 4^{2} = (3⋅4)^{2} = 144

Quotient rules

a^{ n} / a^{ m} = a^{ n}^{-m}

2^{5} / 2^{3} = 2^{5-3} = 4

a^{ n} / b^{ n} = (a / b)^{ n}

4^{3} / 2^{3} = (4/2)^{3} = 8

Power rules

(b^{n})^{m} = b^{n⋅m}

(2^{3})^{2} = 2^{3⋅2} = 64

_{b}n^{m}_{= b}(n^{m})

_{2}3^{2}_{= 2}(3^{2})_{= 512}

^{m}√(b^{n}) =
b^{n/m}

^{2}√(2^{6}) = 2^{6/2} = 8

b^{1/n} = ^{n}√b

8^{1/3} = ^{3}√8 = 2

Negative exponents

b^{-n} = 1 / b^{n}

2^{-3} = 1/2^{3} = 0.125

Metric Prefixes and Conversions

In addition to the use of scientific notation, you will also see a variety of metric prefixes used to describe wavelength, frequency and energy.

Table of Metric Prefixes

In order to do the calculations for this lab we will need to convert frequency and wavelength units to base units of Hertz and Meters. Let's look at some examples.

Example: Convert 610 nanometers to meters First look at the table above to determine the conversion, in this case we see that 1 nanometer is equal to 10^{-9} meters. We simply multiply 610 by 10^{-9}

For more detailed examples, watch the below video.