Section 6.3 Scientific Notation
Subsection 6.3.1 The Basics of Scientific Notation
An October 3, 2016 CBS News headlineβ1β read:The article also later states:Federal Debt in FY 2016 Jumped βthat's Per Household.
When presented in this format, trying to comprehend the value of these numbers can be overwhelming. More commonly, such numbers would be presented in a descriptive manner:By the close of business on Sept. 30, 2016, the last day of fiscal 2016, it had climbed to
The federal debt climbed by trillion dollars in 2016.
The federal debt was trillion dollars at the close of business on Sept. 30, 2016.
Number | US English Name | Power of |
one | ||
ten | ||
hundred | ||
one thousand | ||
ten thousand | ||
one hundred thousand | ||
one million | ||
one billion |
Checkpoint 6.3.2.
Definition 6.3.3.
A positive number is written in scientific notation when it has the form where is an integer and In other words, has precisely one digit to the left of the decimal place. The exponent used here is called the number's order of magnitude. The number is sometimes called the significand or the mantissa.
Some conventions do not require to be between and excluding both values, but that is the convention used in this book.
Subsection 6.3.2 Scientific Notation for Large Numbers
To write a numbers larger than in scientific notation, we write a decimal point after the first non-zero digit of the number and then count the number of places between where the decimal point originally was and where it now is. Scientific notation communicates the size of a number and the order of magnitude just as quickly, but with no need to write long strings of zeros or to try to decipher the language of quintillions, sextillions, etc.Example 6.3.4.
To get a sense of how scientific notation works, let's consider familiar lengths of time converted to seconds.
Length of Time | Length in Seconds | Scientific Notation |
one second | 1 second | second |
one minute | 60 seconds | seconds |
one hour | 3600 seconds | seconds |
one month | 2,628,000 seconds | seconds |
ten years | 315,400,000 seconds | seconds |
79 years (about a lifetime) | 2,491,000,000 seconds | seconds |
Checkpoint 6.3.5.
Checkpoint 6.3.6.
Subsection 6.3.3 Scientific Notation for Small Numbers
Scientific notation can also be useful when working with numbers smaller than As we saw in Table 6.3.1, we can denote thousands, millions, billions, trillions, etc., with positive integer exponents on We can similarly denote numbers smaller than (which are written as tenths, hundredths, thousandths, millionths, billionths, trillionths, etc.), with negative integer exponents on This relationship is outlined in Table 6.3.7.Number | English Name | Power of |
one | ||
one tenth | ||
one hundredth | ||
one thousandth | ||
one ten thousandth | ||
one hundred thousandth | ||
one millionth | ||
one billionth |
Example 6.3.8.
In quantum mechanics, there is an important value called the Planck Constantβ3β. Written as a decimal, the value of the Planck constant (rounded to 4 significant digits) is
In scientific notation, this number will be To determine the exponent, we need to count the number of places from where the decimal is when the number is written as
to where it will be when written in scientific notation:
As a result, in scientific notation, the Planck Constant value is It will be much easier to use in a calculation, and an added benefit is that scientific notation quickly communicates both the value and the order of magnitude of the Planck constant.
Checkpoint 6.3.9.
Checkpoint 6.3.10.
Checkpoint 6.3.11.
Subsection 6.3.4 Multiplying and Dividing Using Scientific Notation
One main reason for having scientific notation is to make calculations involving immensely large or small numbers easier to perform. By having the order of magnitude separated out in scientific notation, we can separate any calculation into two components.Example 6.3.12.
On Sept. 30th, 2016, the US federal debt was about and the US population was about What was the average debt per person that day?
Calculate the answer using the numbers provided, which are not in scientific notation.
First, confirm that the given values in scientific notation are and Then calculate the answer using scientific notation.
We've been asked to answer the same question, but to perform the calculation using two different approaches. In both cases, we'll need to divide the debt by the population.
-
We may need to be working a calculator to handle such large numbers and we have to be careful that we type the correct number of zeros.
\begin{gather*} \frac{19600000000000}{323000000}\approx 60681.11 \end{gather*} -
To perform this calculation using scientific notation, our work would begin by setting up the quotient \(\frac{1.96 \times 10^{13}}{3.23 \times 10^8}\text{.}\) Dividing this quotient follows the same process we did with variable expressions of the same format, such as \(\frac{1.96 w^{13}}{3.23 w^8}\text{.}\) In both situations, we'll divide the coefficients and then use exponent rules to simplify the powers.
\begin{align*} \frac{1.96 \times 10^{13}}{3.23 \times 10^8} \amp= \frac{1.96 }{3.23} \times\frac{10^{13}}{ 10^8}\\ \amp\approx 0.6068111 \times 10^5\\ \amp\approx 60681.11 \end{align*}
The federal debt per capita in the US on September 30th, 2016 was about \(\$60{,}681.11\) per person. Both calculations give us the same answer, but the calculation relying upon scientific notation has less room for error and allows us to perform the calculation as two smaller steps.
Example 6.3.13.
Multiply
Divide
We will simplify the significand/mantissa parts as one step and then simplify the powers of \(10\) as a separate step.
- \begin{align*} \left( 2\times 10^5 \right)\left( 3\times10^4 \right) \amp= \left( 2\times 3 \right)\times \left(10^5 \times 10^4 \right)\\ \amp= 6 \times 10^{9} \end{align*}
- \begin{align*} \frac{8 \times 10^{17}}{4\times 10^2} \amp= \frac{8}{4} \times \frac{10^{17}}{10^2}\\ \amp= 2 \times 10^{15} \end{align*}
While this is a correct value, it is not written using scientific notation. One way to covert this answer into scientific notation is to turn just the coefficient into scientific notation and momentarily ignore the power of ten:
Example 6.3.14.
Multiply or divide as indicated. Write your answer using scientific notation.
Again, we'll separate out the work for the significand/mantissa from the work for the powers of ten. If the resulting coefficient is not between \(1\) and \(10\text{,}\) we'll need to adjust that coefficient to put it into scientific notation.
-
\begin{align*} \left( 8 \times 10^{21} \right)\left( 2 \times 10^{-7} \right) \amp= \left( 8 \times 2 \right)\times\left( 10^{21} \times 10^{-7} \right)\\ \amp= \highlight{16} \times 10^{14}\\ \amp= \highlight{1.6\times 10^1} \times 10^{14}\\ \amp= 1.6 \times 10^{15} \end{align*}
We need to remember to apply the product rule for exponents to the powers of ten.
- \begin{align*} \frac{ 2 \times 10^{-6} }{ 8 \times 10^{-19} } \amp= \frac{ 2 }{ 8 }\times\frac{ 10^{-6} }{ 10^{-19} }\\ \amp= \highlight{0.25} \times 10^{13}\\ \amp= \highlight{2.5\times 10^{-1}} \times 10^{13}\\ \amp= 2.5 \times 10^{12} \end{align*}
Exercises 6.3.5 Exercises
Converting To and From Scientific Notation
1.
Write the following number in scientific notation.
2.
Write the following number in scientific notation.
3.
Write the following number in scientific notation.
4.
Write the following number in scientific notation.
5.
Write the following number in scientific notation.
6.
Write the following number in scientific notation.
7.
Write the following number in scientific notation.
8.
Write the following number in scientific notation.
9.
Write the following number in decimal notation without using exponents.
10.
Write the following number in decimal notation without using exponents.
11.
Write the following number in decimal notation without using exponents.
12.
Write the following number in decimal notation without using exponents.
13.
Write the following number in decimal notation without using exponents.
14.
Write the following number in decimal notation without using exponents.
15.
Write the following number in decimal notation without using exponents.
16.
Write the following number in decimal notation without using exponents.
17.
Write the following number in decimal notation without using exponents.
18.
Write the following number in decimal notation without using exponents.
Arithmetic with Scientific Notation
19.
Multiply the following numbers, writing your answer in scientific notation.
20.
Multiply the following numbers, writing your answer in scientific notation.
21.
Multiply the following numbers, writing your answer in scientific notation.
22.
Multiply the following numbers, writing your answer in scientific notation.
23.
Multiply the following numbers, writing your answer in scientific notation.
24.
Multiply the following numbers, writing your answer in scientific notation.
25.
Divide the following numbers, writing your answer in scientific notation.
26.
Divide the following numbers, writing your answer in scientific notation.
27.
Divide the following numbers, writing your answer in scientific notation.
28.
Divide the following numbers, writing your answer in scientific notation.
29.
Divide the following numbers, writing your answer in scientific notation.
30.
Divide the following numbers, writing your answer in scientific notation.
31.
Divide the following numbers, writing your answer in scientific notation.
32.
Divide the following numbers, writing your answer in scientific notation.
33.
Divide the following numbers, writing your answer in scientific notation.
34.
Divide the following numbers, writing your answer in scientific notation.
35.
Divide the following numbers, writing your answer in scientific notation.
36.
Divide the following numbers, writing your answer in scientific notation.
37.
Simplify the following expression, writing your answer in scientific notation.
38.
Simplify the following expression, writing your answer in scientific notation.
39.
Simplify the following expression, writing your answer in scientific notation.
40.
Simplify the following expression, writing your answer in scientific notation.
41.
Simplify the following expression, writing your answer in scientific notation.
42.
Simplify the following expression, writing your answer in scientific notation.