2.6: Scientific Notation

Converting Floating Decimal Point Numbers to Scientific Notation

A closer look at scientific notation shows that the number is being represented by two parts, the pre-exponential and the exponential (10 to a power). To convert a floating decimal point number to scientific notation you need to multiply or divide the pre-exponential by 10 to a power, until there is just one digit to the left of the decimal point. Because this changes the value of the number, you need to divide or multiply that pre-exponential by 10 to a power (the second term of the notation, noting that division is represented by 10 to a negative exponent), so that the combined expression equals the value of the floating decimal point number.

This is easy to see by looking at the solutions to the next two examples, where you convert a fraction and a number greater than (or equal to) 10 to scientific notation.

Convert the following floating numbers to scientific notation

 

Exercise \(\PageIndex{3}\)

0.00456 (A number smaller than 1)

Answer

Step 1: Since this number is less than 1 you need identify the power of 10 which you can multiply it by and set one digit to the left of the decimal. Here, we can multiply by 1000 or 10³ and force the original number to have one digit to the left of the decimal, but we have changed its value.

0.00456 x 10³ = 4.56 (note, which is not equal to the original number)

Step 2: Since we multiplied it by 10³ we must also divide by 10³ (effectively multiplying the original number by 1 and not changing it's value). Than we express the factor in the denominator as a power of 10.

\[\left ( 0.00456 \right )= \left ( 0.00456 \right )\left ( \frac{10^{3}}{10^{3}} \right )= \left ( \frac{0.00456\times 10^{3}}{10^{3}} \right )= \left ( \frac{4.56}{10^{3}} \right )= 4.56\times 10^{-3}\]

Note, once you get the hang of this you can just count the number of digits you need to move the decimal to bring the original number to one digit to the left of the decimal and multiply that number by 10 to the negative value of those digits. Here we moved the decimal three positions to the right, so we multiply by 10^-3 (ie., divide by 10³).

Exercise \(\PageIndex{4}\)

456.00, (A number larger than or equal to 10)

Answer

Since this number has more than one digit to the left of the decimal, we must divide it by a factor of 10 to make it smaller so that one digit is to the left of the decimal. Then we must multiply everything by 10 to the same power that we divided by 10 in the first step.

\[\left ( 456.00 \right )= \left ( 456.00 \right )\left ( \frac{10^{2}}{10^{2}} \right )= \left ( \frac{456.00}{10^{2}} \right )\times 10^{2}= 4.56\times 10^{2}\]

Note, once you get the hang of this you can just count the number of digits you need to move the decimal to bring the original number to one digit to the left of the decimal and multiply that number by 10 to the value of those digits. Here we moved the decimal two positions to the left, making it 100 times smaller, so we multiply by 10² to maintain the same value.

Significant Digits:

How can you express the number 400 to 2 significant digits?

You must use scientific notation.

4.0 x 10²

As scientific notation always has a decimal point there is never a problem expressing significant digits.

 

Multiplication and Division

Try plugging the following number into your calculator. Odds are you will get an error message, and you need to know how to solve calculations that are bigger or smaller than your calculator can handle.

\[\frac{\left ( 9.47\times 10^{598} \right )\left ( 7.39\times 10^{-98} \right )}{\left ( 1.432\times 10^{-645} \right )\left ( 2.46\times 10^{4} \right )}\]

The trick is to break the above equation into two parts, one dealing with the pre-exponential, and one dealing with the powers of 10. The following video shows how to do this, and the above problem is solved in the exercise \(\PageIndex{5}\) below the video.

Example \(\PageIndex{1}\): multiplication and division with scientific notation

Solve the following problem

\[\frac{\left ( 3.00\times 10^{480} \right )\left ( 4.0\times 10^{-20} \right )}{\left ( 2.00\times 10^{-500} \right )\left ( 1.50\times 10^{70} \right )}\nonumber\]

Solution

Video \(\PageIndex{1}\): Scientific notation in multiplication and division. https://youtu.be/nIFIZkkNRAk

Ans=4.0x10⁸⁹⁰

 

Deeper Look into the Basics

Why do you add exponents when you multiply and subtract when you divide?  A good way to understand this is to look at some simple operations you can do off your head. Like what is 100 times a 1000, or 1000 divided by 100, noting 100 = 10² and 1000 = 10³.

a. One hundred times one thousand

\[100\times 1000= 100,000 \\ \; \\ 10^2\times10^3=10^5 \\ \; \\ 10^{(2+3)}=10^5\]

b. one thousand divided by 100

\[\frac{1000}{100}=10 \\ \; \\ \frac{10^3}{10^2}=10 \\ \; \\ 10^3 \times 10^{-2}=10 \\ \; \\ 10^{(3-2)}=10\\ \; \\ 10^1=10 \]

 

Exercise \(\PageIndex{5}\)

Solve the following problem.

\[\frac{\left ( 9.47\times 10^{598} \right )\left ( 7.39\times 10^{-98} \right )}{\left ( 1.432\times 10^{-645} \right )\left ( 2.46\times 10^{4} \right )} \nonumber \]

Answer

\[= \left ( \frac{9.47x7.39}{1.432x2.46} \right )\left ( \frac{10^{598}\times 10^{-98}}{10^{-645}\times 10^{4}} \right ) \nonumber\]

\[= \left ( 19.86\times 10^{\left ( 598-98+645-4 \right )} \right )\nonumber \]

\[= \left ( 19.9\times 10^{1141} \right )= \left ( 1.99\times 10^{1142} \right )\nonumber \]

 

Closer 

Addition and Subtraction

This can easily be solved with your calculator, the challenge is how do you determine the number of significant Figures in the answer. To do this you need to line them up by multiplying everything by the same power of 10.

Consider the following problem

4.860 x 10¹² + 9.7 x 10¹⁰ + 3.68x10¹¹

Would you believe the answer has four significant Figures? (The solution to this problem is below the following video.)

Example \(\PageIndex{2}\): Addition and subtraction with scientific notation

Solve the following problem and report answer to the correct number of significant Figures:

3 x 10²³ + 3.00 x 10²⁵

Solution

Video \(\PageIndex{2}\): Scientific Notation in addition and subtraction. https://youtu.be/Ghe32OnScdI

(3.00 x 10²⁵)

Exercise \(\PageIndex{4}\)

4.860 x 10¹² + 9.7 x 10¹⁰ + 3.68x10¹¹

Answer

You first need to express all numbers to the same power so you can line up the decimal point. It is suggested that you choose the largest power and make everything else a fraction.

4.860 x 10¹²

+0.097 x 10¹²

+0.368 x 10¹²

5.325 x 10¹²

 

Test Yourself

Homework: Section 2.6

Graded Assignment: Section 2.6

 

Activity \(\PageIndex{1}\)

NOTE: For worked out Solutions of Activity \(\PageIndex{2}\) go to homework section 2.7 The next activity deals with addition and subtraction of numbers in scientific notation.

Activity \(\PageIndex{2}\)

 

Contributors and Attributions