ORCCA Scientific Notation

Section 8.1 Scientific Notation

Objectives: PCC Course Content and Outcome Guide

Very large and very small numbers can be awkward to write and calculate with. These kinds of numbers can show in the sciences. For example in biology, a human hair might be as thick as

0.000181

meters. And the closest that Mars gets to the sun is

206620000

meters. Keeping track of the decimal places and extra zeros raises the potential for mistakes to be made. In this section, we discuss a format used for very large and very small numbers called scientific notation that helps alleviate the issues with these numbers.

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Figure 8.1.1. Alternative Video Lesson

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Subsection 8.1.1 The Basics of Scientific Notation

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An October 3, 2016 CBS News headline

http://www.cnsnews.com/news/article/terence-p-jeffrey/federal-debt-fy-2016-jumped-142282704745246

read:

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Federal Debt in FY 2016 Jumped $$ 1,422,827,047,452.46$ —that’s $$ 12,036$ Per Household.

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The article also later states:

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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 .$

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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.

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In science, government, business, and many other disciplines, it’s not uncommon to deal with very large numbers like these. When numbers get this large, it can be hard to discern when a number has eleven digits and when it has twelve.

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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:

Number	US English Name	Power of $10$
$1$	one	$10^{0}$
$10$	ten	$10^{1}$
$100$	hundred	$10^{2}$
$1,000$	one thousand	$10^{3}$
$10,000$	ten thousand	$10^{4}$
$100,000$	one hundred thousand	$10^{5}$
$1,000,000$	one million	$10^{6}$
$1,000,000,000$	one billion	$10^{9}$

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Figure 8.1.2. Whole Number Powers of

10

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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.

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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

imagine.gsfc.nasa.gov/features/cosmic/nearest_star_info.html

. Proxima Centauri is about

25,000,000,000,000

miles from our sun. Again, many will find the descriptive language easier to read: Proxima Centauri is about

25

trillion miles from our sun.

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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. So first, let’s take a moment to review that level of basic arithmetic.

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Checkpoint 8.1.3.

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Perform the following operations:

Multiply $5.7$ by $10 .$
Multiply $3.1$ by $10000 .$

Explanation.

$5.7 \times 10 = 57$

$10 = 10^{1}$ and multiplying by $10^{1}$ moved the decimal point one place to the right.
$3.1 \times 10000 = 31000$

$10000 = 10^{4}$ and multiplying by $10^{4}$ moved the decimal point four places to the right.

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Multiplying a number by

10^{n}

where

n

is a positive integer had the effect of moving the decimal point

n

places to the right.

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Every number can be written as a product of a number between

1

and

10

and a power of

10 .

For example,

650 = 6.5 \times 100 .

Since

100 = 10^{2},

we can also write

650 = 6.5 \times 10^{2}

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

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Definition 8.1.4.

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A positive number is written in scientific notation when it has the form

a \times 10^{n}

where

n

is an integer and

1 \leq a < 10 .

In other words,

a

has precisely one nonzero 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.

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Some conventions do not require

a

to be between

1

and

10,

excluding both values, but that is the convention used in this book.

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Some calculators and computer readouts cannot display exponents in superscript. In some cases, these devices will display scientific notation in the form 6.5E2 instead of

6.5 \times 10^{2} .

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Subsection 8.1.2 Scientific Notation for Large Numbers

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To write a number larger than

10

in scientific notation, like

89412,

first write the number with the decimal point right after its first digit, like

8.9412 .

Now count how many places there are between where the decimal point originally was and where it is now.

8. \overset{4}{\overset{⏞}{9412}}

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Use that count as the power of

10 .

In this example, we have

89412 = 8.9412 \times 10^{4}

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Scientific notation communicates the “essence” of the number (

8.9412

) and then its size, or order of magnitude (

10^{4}

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Example 8.1.5.

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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	$1 \times 10^{0}$ second
one minute	60 seconds	$6 \times 10^{1}$ seconds
one hour	3600 seconds	$3.6 \times 10^{3}$ seconds
one month	2,628,000 seconds	$2.628 \times 10^{6}$ seconds
ten years	315,400,000 seconds	$3.154 \times 10^{8}$ seconds
79 years (about a lifetime)	2,491,000,000 seconds	$2.491 \times 10^{9}$ seconds

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Note that roughly

2.6

million seconds is one month, while roughly

2.5

billion seconds is an entire lifetime.

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Checkpoint 8.1.6.

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Write each of the following in scientific notation.

The federal debt at the close of business on Sept. 30, 2016: about $19,600,000,000,000$ dollars.
The world’s population
³
www.prb.org/pdf16/prb-wpds2016-web-2016.pdf
in 2016: about $7,418,000,000$ people.

Explanation.

To convert the federal debt to scientific notation, we will count the number of digits after the first nonzero digit (which happens to be a $1$ here). Since there are $13$ places after the first nonzero digit, we write:

$1 \overset{13 places}{\overset{⏞}{9,600,000,000,000}} dollars = 1.96 \times 10^{13} dollars$
Since there are nine places after the first nonzero digit of $7,$ the world’s population in 2016 was about

$7, \overset{9 places}{\overset{⏞}{418,000,000}} people = 7.418 \times 10^{9} people$

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Checkpoint 8.1.7.

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Convert each of the following from scientific notation to decimal notation (without any exponents).

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The earth’s diameter is about $1.27 \times 10^{7}$ meters.
As of 2019, there are $3.14 \times 10^{13}$ known digits of $π .$

Explanation.

To convert this number to decimal notation we will move the decimal point after the digit $1$ seven places to the right, including zeros where necessary. The earth’s diameter is:

$1.27 \times 10^{7} meters = 1 \overset{7 places}{\overset{⏞}{2,700,000}} meters .$
As of 2019 there are

$3.14 \times 10^{13} = 3 \overset{13 places}{\overset{⏞}{1,400,000,000,000}}$

known digits of $π .$

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Subsection 8.1.3 Scientific Notation for Small Numbers

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Scientific notation can also be useful when working with numbers smaller than

1 .

As we saw in Figure 2, we can represent thousands, millions, billions, trillions, etc., with positive integer exponents on

10 .

We can similarly represent 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 Figure 8.

Number	English Name	Power of $10$
$1$	one	$10^{0}$
$0.1$	one tenth	$\frac{1}{10} = 10^{- 1}$
$0.01$	one hundredth	$\frac{1}{100} = 10^{- 2}$
$0.001$	one thousandth	$\frac{1}{1,000} = 10^{- 3}$
$0.0001$	one ten thousandth	$\frac{1}{10,000} = 10^{- 4}$
$0.00001$	one hundred thousandth	$\frac{1}{100,000} = 10^{- 5}$
$0.000001$	one millionth	$\frac{1}{1,000,000} = 10^{- 6}$
$0.000000001$	one billionth	$\frac{1}{1,000,000,000} = 10^{- 9}$

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Figure 8.1.8. Negative Integer Powers of

10

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To see how this works with a digit other than

1,

let’s look at

0.005 .

When we state

0.005

as a number, we say “5 thousandths.” Thus

0.005 = 5 \times \frac{1}{1000} .

The fraction

\frac{1}{1000}

can be written as

\frac{1}{10^{3}},

which we know is equivalent to

10^{- 3} .

Using negative exponents, we can then rewrite

0.005

5 \times 10^{- 3} .

This is the scientific notation for

0.005 .

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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 nonzero 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

0.005

to scientific notation, we have:

0 \overset{3}{\overset{⏞}{. 005}} = 5 \times 10^{- 3}

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Example 8.1.9.

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In quantum mechanics, there is an important value called Planck’s Constant

⁴

en.wikipedia.org/wiki/Planck_constant

. Written as a decimal, the value of Planck’s constant (rounded to six significant digits) is

0.000 000 000 000 000 000 000 000 000 000 000 662 607 .

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In scientific notation, this number will be

6.62607 \times 10^{?} .

To determine the exponent, we need to count the number of places from where the decimal originally is to where we will move it (following the first “6”):

0 \overset{34 places}{\overset{⏞}{. 000 000 000 000 000 000 000 000 000 000 000 6}} 62 607

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So in scientific notation, Planck’s Constant is

6.62607 \times 10^{- 34} .

It will be much easier to use

6.62607 \times 10^{- 34}

in a calculation, and an added benefit is that scientific notation quickly communicates both the value and the order of magnitude of Planck’s Constant.

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Checkpoint 8.1.10.

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Write each of the following in scientific notation.

The weight of a single grain of long grain rice is about $0.029$ grams.
The gate pitch of a microprocessor
⁵
www.prb.org/pdf16/prb-wpds2016-web-2016.pdf
is $0.000 000 014$ meters

Explanation.

To convert this weight to scientific notation, we must first move the decimal behind the first nonzero digit to obtain $2.9,$ which requires that we move the decimal point 2 places. Thus we have:

$0 \overset{2}{\overset{⏞}{.02}} 9 grams = 2.9 \times 10^{- 2} grams$
The gate pitch of a microprocessor is:

$0 \overset{8 places}{\overset{⏞}{.000 000 01}} 4 meters = 1.4 \times 10^{- 8} meters$

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Checkpoint 8.1.11.

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Convert each of the following from scientific notation to decimal notation (without any exponents).

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A download speed of $7.53 \times 10^{- 3}$ Gigabyte per second.
The weight of a poppy seed is about $3 \times 10^{- 7}$ kilograms

Explanation.

To convert a download speed of $7.53 \times 10^{- 3}$ Gigabyte per second to decimal notation, we will move the decimal point $3$ places to the left and include the appropriate number of zeros:

$7.53 \times 10^{- 3} Gigabyte per second = 0 \overset{3}{\overset{⏞}{.007}} 53 Gigabyte per second$
The weight of a poppy seed is about:

$3 \times 10^{- 7} kilograms = 0 \overset{7 places}{\overset{⏞}{.0000003}} kilograms$

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Checkpoint 8.1.12.

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Decide if the numbers are written in scientific notation or not. Use Definition 4.

The number $7 \times 10^{1.9}$
- ?
- is
- is not
in scientific notation.
The number $2.6 \times 10^{- 31}$
- ?
- is
- is not
in scientific notation.
The number $10 \times 7^{4}$
- ?
- is
- is not
in scientific notation.
The number $0.93 \times 10^{3}$
- ?
- is
- is not
in scientific notation.
The number $4.2 \times 10^{0}$
- ?
- is
- is not
in scientific notation.
The number $12.5 \times 10^{- 6}$
- ?
- is
- is not
in scientific notation.

Explanation.

The number $7 \times 10^{1.9}$ is not in scientific notation. The exponent on the 10 is required to be an integer and $1.9$ is not.
The number $2.6 \times 10^{- 31}$ is in scientific notation.
The number $10 \times 7^{4}$ is not in scientific notation. The base must be $10,$ not $7 .$
The number $0.93 \times 10^{3}$ is not in scientific notation. The coefficient of the 10 must be between $1$ (inclusive) and $10 .$
The number $4.2 \times 10^{0}$ is in scientific notation.
The number $12.5 \times 10^{- 6}$ is not in scientific notation. The coefficient of the 10 must be between $1$ (inclusive) and $10 .$

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Subsection 8.1.4 Multiplying and Dividing Using Scientific Notation

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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.

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Example 8.1.13.

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On Sept. 30th, 2016, the US federal debt

⁶

www.census.gov/popclock

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?

Calculate the answer using the numbers provided, which are not in scientific notation.
First, confirm that the given values in scientific notation are $1.96 \times 10^{13}$ and $3.23 \times 10^{8} .$ 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.

We may need to use a calculator to handle such large numbers and we have to be careful that we type the correct number of 0s.

$\begin{matrix} \frac{19600000000000}{323000000} \approx 60681.11 \end{matrix}$
To perform this calculation using scientific notation, our work would begin by setting up the quotient as $\frac{1.96 \times 10^{13}}{3.23 \times 10^{8}} .$ Dividing this quotient follows the same process we did with variable expressions of the same format, such as $\frac{1.96 w^{13}}{3.23 w^{8}} .$ In both situations, we’ll divide the coefficients and then use exponent properties to simplify the powers.

$\begin{aligned} \frac{1.96 \times 10^{13}}{3.23 \times 10^{8}} & = \frac{1.96}{3.23} \times \frac{10^{13}}{10^{8}} \\ \approx 0.6068111 \times 10^{5} \\ \approx 60681.11 \end{aligned}$

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.

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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 properties.

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Example 8.1.14.

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Multiply $(2 \times 10^{5}) (3 \times 10^{4}) .$
Divide $\frac{8 \times 10^{17}}{4 \times 10^{2}} .$

Explanation.

We will simplify the significand/mantissa parts as one step and then simplify the powers of

10

as a separate step.

$\begin{aligned} (2 \times 10^{5}) (3 \times 10^{4}) & = (2 \times 3) \times (10^{5} \times 10^{4}) \\ = 6 \times 10^{9} \end{aligned}$
$\begin{aligned} \frac{8 \times 10^{17}}{4 \times 10^{2}} & = \frac{8}{4} \times \frac{10^{17}}{10^{2}} \\ = 2 \times 10^{15} \end{aligned}$

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Often when we multiply or divide numbers in scientific notation, the resulting value will not be in scientific notation. Suppose we were multiplying

(9.3 \times 10^{17}) (8.2 \times 10^{- 6})

and need to state our answer using scientific notation. We would start as we have previously:

\begin{aligned} (9.3 \times 10^{17}) (8.2 \times 10^{- 6}) & = (9.3 \times 8.2) \times (10^{17} \times 10^{- 6}) \\ = 76.26 \times 10^{11} \end{aligned}

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While this is a correct value, it is not written using scientific notation. One way to convert this answer into scientific notation is to turn just the coefficient into scientific notation and momentarily ignore the power of ten:

\begin{aligned} = 76.26 \times 10^{11} \\ = 7.626 \times 10^{1} \times 10^{11} \end{aligned}

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

\begin{aligned} = 7.626 \times 10^{1} \times 10^{11} \\ = 7.626 \times 10^{12} \end{aligned}

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Example 8.1.15.

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

$(8 \times 10^{21}) (2 \times 10^{- 7})$
$\frac{2 \times 10^{- 6}}{8 \times 10^{- 19}}$

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,

we’ll need to adjust that coefficient to put it into scientific notation.

$\begin{aligned} (8 \times 10^{21}) (2 \times 10^{- 7}) & = (8 \times 2) \times (10^{21} \times 10^{- 7}) \\ = 16 \times 10^{14} \\ = 1.6 \times 10^{1} \times 10^{14} \\ = 1.6 \times 10^{15} \end{aligned}$

We need to remember to apply the product property for exponents to the powers of ten.
$\begin{aligned} \frac{2 \times 10^{- 6}}{8 \times 10^{- 19}} & = \frac{2}{8} \times \frac{10^{- 6}}{10^{- 19}} \\ = 0.25 \times 10^{13} \\ = 2.5 \times 10^{- 1} \times 10^{13} \\ = 2.5 \times 10^{12} \end{aligned}$

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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

3 \times 10^{7}

feet. To perform this calculation, we first remember the formula for the area of a square,

A = s^{2}

and then substitute

3 \times 10^{7}

for

s :

A = {(3 \times 10^{7})}^{2} .

To perform this calculation, we’ll need to remember to use the product to a power property and the power to a power property:

\begin{aligned} A & = {(3 \times 10^{7})}^{2} \\ = {(3)}^{2} \times {(10^{7})}^{2} \\ = 9 \times 10^{14} \end{aligned}

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Reading Questions 8.1.5 Reading Questions

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1.

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Which number is very large and which number is very small?

9.99 \times 10^{- 47} 1.01 \times 10^{23}

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2.

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Since some computer/calculator screens can’t display an exponent, how might a computer/calculator display the number

2.318 \times 10^{13} ?

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3.

🔗

Why do we bother having scientific notation for numbers?

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Write the following number in scientific notation.

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270000 =

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2.

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Write the following number in scientific notation.

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3700 =

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3.

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Write the following number in scientific notation.

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470 =

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4.

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Write the following number in scientific notation.

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57000 =

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5.

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Write the following number in scientific notation.

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0.0067 =

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6.

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Write the following number in scientific notation.

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0.00077 =

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7.

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Write the following number in scientific notation.

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0.087 =

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8.

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Write the following number in scientific notation.

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0.0097 =

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Exercise Group.

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9.

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Write the following number in decimal notation without using exponents.

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1.7 \times 10^{5} =

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10.

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Write the following number in decimal notation without using exponents.

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2.7 \times 10^{4} =

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11.

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Write the following number in decimal notation without using exponents.

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3.69 \times 10^{3} =

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12.

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Write the following number in decimal notation without using exponents.

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4.69 \times 10^{5} =

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13.

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Write the following number in decimal notation without using exponents.

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5.69 \times 10^{0} =

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14.

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Write the following number in decimal notation without using exponents.

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6.68 \times 10^{0} =

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15.

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Write the following number in decimal notation without using exponents.

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7.7 \times 10^{- 2} =

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16.

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Write the following number in decimal notation without using exponents.

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8.6 \times 10^{- 3} =

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17.

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Write the following number in decimal notation without using exponents.

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9.67 \times 10^{- 4} =

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18.

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Write the following number in decimal notation without using exponents.

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1.67 \times 10^{- 2} =

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Arithmetic with Scientific Notation.

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19.

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

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(3 \times 10^{2}) (6 \times 10^{3}) =

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20.

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

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(4 \times 10^{4}) (3 \times 10^{2}) =

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21.

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

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(5 \times 10^{2}) (8 \times 10^{5}) =

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22.

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

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(6 \times 10^{3}) (5 \times 10^{4}) =

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23.

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

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(7 \times 10^{5}) (2 \times 10^{3}) =

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24.

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

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(7 \times 10^{3}) (8 \times 10^{5}) =

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