Skip to main content

Section 6.3 Scientific Notation

Having just learned more about exponents, including negative exponents, we can discuss a format used for very large and very small numbers called scientific notation.

Subsection 6.3.1 The Basics of Scientific Notation

An October 3, 2016 CBS News headline 1  read:

Federal Debt in FY 2016 Jumped $1,422,827,047,452.46β€”that's $12,036 Per Household.

The article also later states:

By the close of business on Sept. 30, 2016, the last day of fiscal 2016, it had climbed to $19,573,444,713,936.79.

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:

  • The federal debt climbed by 1.42 trillion dollars in 2016.

  • The federal debt was 19.6 trillion dollars at the close of business on Sept. 30, 2016.

Unless we're presented with such news items, most of us deal with numbers no larger than the thousands in our daily life. In science, government, business, and many other disciplines, it's not uncommon to deal with much larger numbers. When numbers get this large, it can be hard to distinguish between a number that has nine or twelve digits. On the other hand, we have descriptive language that allows us grasp the value and not be lost in the sheer size of the number.

We have descriptive language for all numbers, based on the place value of the different digits: ones, tens, thousands, ten thousands, etc. We tend to rely upon this language more when we start dealing with larger numbers. Here's a chart for some of the most common numbers we see and use in the world around us:

Table 6.3.1. Whole Number Powers of 10
Number US English Name Power of 10
1 one 100
10 ten 101
100 hundred 102
1,000 one thousand 103
10,000 ten thousand 104
100,000 one hundred thousand 105
1,000,000 one million 106
1,000,000,000 one billion 109

Each number above has a corresponding power of ten and this power of ten will be important as we start to work with the content in this section.

This descriptive language also covers even larger numbers: trillion, quadrillion, quintillion, sextillion, septillion, and so on. There's also corresponding language to describe very small numbers, such as thousandth, millionth, billionth, trillionth, etc.

Through centuries of scientific progress, humanity became increasingly aware of very large numbers and very small measurements. As one example, the star that is nearest to our sun is Proxima Centauri 2 . Proxima Centauri is about 25,000,000,000,000 miles from our sun. Again, many will find the descriptive language easier to digest: Proxima Centauri is about 25 trillion miles from our sun.

To make computations involving such numbers more manageable, a standardized notation called scientific notation was established. The foundation of scientific notation is the fact that multiplying or dividing by a power of 10 will move the decimal point of a number so many places to the right or left, respectively.

Checkpoint 6.3.2.

Multiplying a number by 10n where n is a positive integer had the effect of moving the decimal point n places to the right.

Every number can be written as a product of a number between 1 and 10 and a power of 10. For example, 650=6.5Γ—100. Since 100=102, we can also write

650=6.5Γ—102

and this is our first example of writing a number in scientific notation.

Definition 6.3.3.

A positive number is written in scientific notation when it has the form aΓ—10n where n is an integer and 1≀a<10. In other words, a has precisely one digit to the left of the decimal place. The exponent n used here is called the number's order of magnitude. The number a is sometimes called the significand or the mantissa.

Some conventions do not require a to be between 1 and 10, 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 10 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 1Γ—100 second
one minute 60 seconds 6Γ—101 seconds
one hour 3600 seconds 3.6Γ—103 seconds
one month 2,628,000 seconds 2.628Γ—106 seconds
ten years 315,400,000 seconds 3.154Γ—108 seconds
79 years (about a lifetime) 2,491,000,000 seconds 2.491Γ—109 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 1. As we saw in Table 6.3.1, we can denote thousands, millions, billions, trillions, etc., with positive integer exponents on 10. We can similarly denote numbers smaller than 1 (which are written as tenths, hundredths, thousandths, millionths, billionths, trillionths, etc.), with negative integer exponents on 10. This relationship is outlined in Table 6.3.7.

Table 6.3.7. Negative Integer Powers of 10
Number English Name Power of 10
1 one 100
0.1 one tenth 110=10βˆ’1
0.01 one hundredth 1100=10βˆ’2
0.001 one thousandth 11,000=10βˆ’3
0.0001 one ten thousandth 110,000=10βˆ’4
0.00001 one hundred thousandth 1100,000=10βˆ’5
0.000001 one millionth 11,000,000=10βˆ’6
0.000000001 one billionth 11,000,000,000=10βˆ’9

To see how this works with a digit other than 1, let's look at 0.05. When we state 0.05 as a number, we say β€œ5 hundredths.” Thus, 0.05=5Γ—1100. The fraction 1100 can be written as 1102, which we know is equivalent to 10βˆ’2. Using negative exponents, we can then rewrite 0.05 as 5Γ—10βˆ’2. This is the scientific notation for 0.05.

In practice, we won't generally do that much computation. To write a small number in scientific notation we start as we did before and place the decimal point behind the first non-zero digit. We then count the number of decimal places between where the decimal had originally been and where it now is. Keep in mind that negative powers of ten are used to help represent very small numbers (smaller than 1) and positive powers of ten are used to represent very large numbers (larger than 1). So to convert 0.05 to scientific notation, we have:

0.05⏞2 places=5Γ—10βˆ’2
Example 6.3.8.

In quantum mechanics, there is an important value called the Planck Constant 3 . Written as a decimal, the value of the Planck constant (rounded to 4 significant digits) is

0.0000000000000000000000000000000006626.

In scientific notation, this number will be 6.626Γ—10?. To determine the exponent, we need to count the number of places from where the decimal is when the number is written as

0.0000000000000000000000000000000006626

to where it will be when written in scientific notation:

0.0000000000000000000000000000000006⏞34 places626

As a result, in scientific notation, the Planck Constant value is 6.626Γ—10βˆ’34. It will be much easier to use 6.626Γ—10βˆ’34 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 $19,600,000,000,000 and the US population was about 323,000,000. What was the average debt per person that day?

  1. Calculate the answer using the numbers provided, which are not in scientific notation.

  2. First, confirm that the given values in scientific notation are 1.96Γ—1013 and 3.23Γ—108. Then calculate the answer using scientific notation.

Explanation

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.

  1. 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*}
  2. 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.

Whenever we multiply or divide numbers that are written in scientific notation, we must separate the calculation for the coefficients from the calculation for the powers of ten, just as we simplified earlier expressions using variables and the exponent rules.

Example 6.3.13.
  1. Multiply (2Γ—105)(3Γ—104).

  2. Divide 8Γ—10174Γ—102.

Explanation

We will simplify the significand/mantissa parts as one step and then simplify the powers of \(10\) as a separate step.

  1. \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*}
  2. \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*}

Often when we multiply or divide numbers in scientific notation, the resulting value will not be in scientific notation. Suppose we were multiplying (9.3Γ—1017)(8.2Γ—10βˆ’6) and need to state our answer using scientific notation. We would start as we have previously:

(9.3Γ—1017)(8.2Γ—10βˆ’6)=(9.3Γ—8.2)Γ—(1017Γ—10βˆ’6)=76.26Γ—1011

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:

=76.26Γ—1011=7.626Γ—101Γ—1011

Now that the coefficient fits into the proper format, we can combine the powers of ten and have our answer written using scientific notation.

=7.626Γ—101Γ—1011=7.626Γ—1012
Example 6.3.14.

Multiply or divide as indicated. Write your answer using scientific notation.

  1. (8Γ—1021)(2Γ—10βˆ’7)
  2. 2Γ—10βˆ’68Γ—10βˆ’19
Explanation

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.

  1. \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.

  2. \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*}

There are times where we will have to raise numbers written in scientific notation to a power. For example, suppose we have to find the area of a square whose radius is 3Γ—107 feet. To perform this calculation, we first remember the formula for the area of a square, A=s2 and then substitute 3Γ—107 for s: A=(3Γ—107)2. To perform this calculation, we'll need to remember to use the product to a power rule and the power to a power rule:

A=(3Γ—107)2=(3)2Γ—(107)2=9Γ—1014

Exercises 6.3.5 Exercises

Converting To and From Scientific Notation
1.

Write the following number in scientific notation.

910000=

2.

Write the following number in scientific notation.

11000=

3.

Write the following number in scientific notation.

2100=

4.

Write the following number in scientific notation.

310000=

5.

Write the following number in scientific notation.

0.0041=

6.

Write the following number in scientific notation.

0.00051=

7.

Write the following number in scientific notation.

0.061=

8.

Write the following number in scientific notation.

0.0071=

9.

Write the following number in decimal notation without using exponents.

8.1Γ—102=

10.

Write the following number in decimal notation without using exponents.

9.1Γ—105=

11.

Write the following number in decimal notation without using exponents.

1.11Γ—103=

12.

Write the following number in decimal notation without using exponents.

2.11Γ—102=

13.

Write the following number in decimal notation without using exponents.

3.1Γ—100=

14.

Write the following number in decimal notation without using exponents.

4.1Γ—100=

15.

Write the following number in decimal notation without using exponents.

5.1Γ—10βˆ’4=

16.

Write the following number in decimal notation without using exponents.

6.1Γ—10βˆ’2=

17.

Write the following number in decimal notation without using exponents.

7.09Γ—10βˆ’3=

18.

Write the following number in decimal notation without using exponents.

8.09Γ—10βˆ’4=

Arithmetic with Scientific Notation
19.

Multiply the following numbers, writing your answer in scientific notation.

(9Γ—104)(7Γ—104)=

20.

Multiply the following numbers, writing your answer in scientific notation.

(2Γ—102)(4Γ—103)=

21.

Multiply the following numbers, writing your answer in scientific notation.

(2Γ—104)(2Γ—102)=

22.

Multiply the following numbers, writing your answer in scientific notation.

(3Γ—105)(7Γ—105)=

23.

Multiply the following numbers, writing your answer in scientific notation.

(4Γ—103)(4Γ—104)=

24.

Multiply the following numbers, writing your answer in scientific notation.

(5Γ—105)(9Γ—103)=

25.

Divide the following numbers, writing your answer in scientific notation.

3.6Γ—1036Γ—102=

26.

Divide the following numbers, writing your answer in scientific notation.

2.8Γ—1057Γ—103=

27.

Divide the following numbers, writing your answer in scientific notation.

7.2Γ—1038Γ—104=

28.

Divide the following numbers, writing your answer in scientific notation.

5.4Γ—1049Γ—102=

29.

Divide the following numbers, writing your answer in scientific notation.

6Γ—1052Γ—10βˆ’2=

30.

Divide the following numbers, writing your answer in scientific notation.

1.8Γ—1032Γ—10βˆ’4=

31.

Divide the following numbers, writing your answer in scientific notation.

1.8Γ—1043Γ—10βˆ’3=

32.

Divide the following numbers, writing your answer in scientific notation.

1.2Γ—1024Γ—10βˆ’5=

33.

Divide the following numbers, writing your answer in scientific notation.

4Γ—10βˆ’35Γ—105=

34.

Divide the following numbers, writing your answer in scientific notation.

3Γ—10βˆ’26Γ—104=

35.

Divide the following numbers, writing your answer in scientific notation.

2.1Γ—10βˆ’47Γ—103=

36.

Divide the following numbers, writing your answer in scientific notation.

6.4Γ—10βˆ’28Γ—105=

37.

Simplify the following expression, writing your answer in scientific notation.

(5Γ—106)2=

38.

Simplify the following expression, writing your answer in scientific notation.

(2Γ—102)3=

39.

Simplify the following expression, writing your answer in scientific notation.

(2Γ—108)2=

40.

Simplify the following expression, writing your answer in scientific notation.

(2Γ—105)3=

41.

Simplify the following expression, writing your answer in scientific notation.

(3Γ—102)4=

42.

Simplify the following expression, writing your answer in scientific notation.

(3Γ—108)2=