Lesson 36 Text - Diablo Valley Collegevoyager.dvc.edu/~lmonth/PreAlg/lesson36student.pdf · scientific notation for the result 670,800,000,000,000. Scientific Notation to Standard

Free Pre-Algebra Lesson 36 ! page 1

© 2010 Cheryl Wilcox

Lesson 36

Scientific Notation

There’s more than one way to write any number. Depending on the circumstances, we might write the quantity “three” as III (Roman numerals), $3.00, or 3.0 cm, or 3 burritos, or “a few” friends. The fractions 2/3, 4/6, 6/9, etc are all equal quantities expressed differently, and we could also write a repeating decimal 0.666… to represent the same amount. Scientific notation is yet another way to write numbers, especially helpful when numbers are very large. You’ve probably seen scientific notation in a science textbook. Your calculator uses scientific notation when the number of digits in an answer exceeds the number of digits in the display.

Example: The radius of the earth is about 4000 miles. What is the approximate volume of the earth?

V =4

3!r

3=

4

3! 4000( )

3

When you enter this in your calculator, you will see something like:

You may wonder why a number you expect to be very large has a decimal point after the ones place. And what is over in the right-hand corner? (Your calculator may only display the exponent 11 instead of the entire expression x 1011.) Because the answer is so large, the calculator has switched automatically to scientific notation. The answer 2.68082572 x 1011 is written in scientific notation. A number written in scientific notation has two parts, a decimal part multiplied by a power of ten:

2.68082572 x 1011 The decimal part is a number between one and ten. In this section, you will learn • to convert numbers written in scientific notation to ordinary place value notation; • to convert ordinary numbers to scientific notation; • some contexts in which scientific notation is helpful; • how we use an adaptation of scientific notation in everyday speech.

Powers of Ten Tens are easy to multiply, because our number system is based on 10s.

So 102 = (10 )(10 )= 100, 103 = (10 )(10 )(10 ) = 1000, 104 = 10,000 and so forth. You can notice quickly that the exponent tells you how many zeros to write after the 1.

In fact, the powers of 10 are just the place value groups: 103 is one thousand, 104 is ten thousand, 106 is one million, etc.

Example: Write the numbers as indicated.

a. Write the number ten trillion as a power of ten.

As the place values increase, we increase the exponent of ten by 1. In the ten trillions place, the exponent will be 13.

Ten trillion = 1013

Write the number 1012 in standard notation and write its name in words.

The exponent 12 is the number of zeros after the digit 1.

1012 = 1,000,000,000,000 = one trillion

trillions billions millions thousands

100 10 1 100 10 1 100 10 1 100 10 1 100 10 1

2 3, 0 7 4, 1 8 1, 3 0 4, 5 2 5


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1

3, 4 8 5, 2 1 8

109 108 107 106 105 104 103 102 101


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1

4 0 9, 0 0 0, 0 0 0



Check the multiplication table below using your calculator. Then read the observations to the right.

! 10

! 10

2

! 10

3

! 10

6

1.0 10 100 1,000 1,000,000 This row is just the powers of ten written in everyday place value notation.

3.0 30 300 3,000 3,000,000 Multiplying by 3 replaces the 1 in the power of ten by a 3.

43.0 430 4300 43,000 43,000,000 Multiplying by 43 replaces the 1 in the power of ten by the 3 in the ones place of 43. The 4 remains in front of the 3.

0.2 02 020 0200 0,200,000 The ghostly 0 that was in the ones place replaces the digit 1 in the power of 10. The 2 that was in the tenths keeps its relative position to the 0.

3.2 32 320 3,200 3,200,000 The 3 in the ones place replaces the digit 1 in the power of 10. The 2 in the tenths place keeps its relative position to the 3.

Example: Multiply 67.08 x 1013 without using a calculator (a calculator cannot display all the digits).

1 Write out the power of ten (1013) in

everyday place value notation. The exponent 13 is the number of zeros after the digit 1 in the place value expression.

2 Line up the ones digit of the

number you are multiplying with the digit 1 in the power of ten. Keep other digits in their same relative positions.

3 Drop the decimal point and fill in

any missing place values with zeros, inserting commas to match the power of ten.

1013

= 10,000,000,000,000

1013 is ten trillion.

1 0,000,000,000,000

67.08

10,000,000,000,000

670,800,000,000,000

67.08 x 1013 = 670,800,000,000,000 The multiplication 67.08 x 1013 is not a number written in scientific notation, because the decimal number 67.08 is not between 1 and 10. If you enter this multiplication on your calculator, you will see the result 6.708 x 1014, which is correct scientific notation for the result 670,800,000,000,000. Scientific Notation to Standard Notation

Example: Write the approximate volume of the earth, 2.68082572 x 1011 cubic miles, in everyday place value notation.

1. Write 1011 out in place value notation. 1 00000000000

2. Align the 2 in the ones place with the digit 1. 2.68082572 3. Drop the decimal point and fill in any empty places with 0s.

2 68082572000

The approximate volume of the earth is 268,082,572,000 cubic miles.

The number is automatically rounded because the calculator can’t display all the digits. However, since the radius of the earth as given originally was rounded to one significant figure, we really ought to write the volume approximation with one significant figure as well: about 300 billion cubic miles. (You are not expected to calculate significant figures in this section.) You can see that converting scientific notation to place value notation is just a multiplication by the power of 10.



Standard Notation to Scientific Notation To change a number written in standard notation to scientific notation, follow the steps below:

Example: Write the number 45,900,000,000,000,000,000 in scientific notation.

1 Write out the number without

commas and put a decimal point after the first digit.

2 Count the number of digits after the

(new) decimal point. This will be the exponent for 10.

3 Drop any zeros to the right of the

last non-zero digit, and write the decimal times the power of 10.

4.5900000000000000000 4.5900000000000000000

19 digits after the decimal point.

x 1019

4.59 x 1019

This procedure always puts the leading digit of any number in the ones place of the decimal part of the scientific notation, resulting in a decimal part that is between 1 and 10.

Example: Write the number 96 billion in scientific notation.

Write 96 billion in standard place value notation. 96,000,000,000

1. Write a decimal point after the first digit and drop the commas. 9.6000000000 2. Count the digits after the decimal point and use that number for the exponent of 10.

1010

3. Drop extra zeros from the right of the number and write as a multiplication with the power of ten.

9.6 x 1010

Everyday Speech Sometimes numbers are written with a decimal point and a place value name. You see this in news articles and financial statements. For example, “That penthouse cost $3.5 million!” Interpret this as 3.5 x 1,000,000 and you see that it means $3,500,000 (three million, five hundred thousand dollars). Unlike true scientific notation, you are not limited to putting the decimal point after the first digit. Instead, use the largest place value group name, and use a decimal point for digits in the group below.

Example: Convert the underlined numbers as requested.

In 2009, a total of 33.3 million pounds of toxic chemicals were released into the air in the state of New York.

Write in standard notation:

33,300,000

The amount of financial aid increased by over $1,500,000.

Write using the largest place value name.

$1.5 million



Why Use Scientific Notation? There are about 25 trillion red blood cells in our body at one time. About 12,500,000,000,000 locusts appeared in Nebraska in 1875. There are about 1018 molecules in a snowflake. (The Sizeasaurus, Strauss, 1995) When working with large numbers, it helps to have a condensed notation to express their size in a way that is easily comparable.

25 trillion red blood cells 2.5 x 1013

12,500,000,000,000 locusts 1.25 x 1013

1018 molecules in a snowflake 1 x 1018

By writing all the numbers in scientific notation, it is much easier to see that the number of red blood cells in your body is close to the same as the the number of locusts that appeared in Nebraska, (Is that significant? creepy? meaningless?) whereas the number of molecules in a snowflake is much greater. Comparing Orders of Magnitude An increase of one place value, called an order of magnitude, is the same as increasing the exponent by 1, say from 1013 to 1014. It makes the number 10 times greater. Since the exponent 18 is five more than the exponent 13, the number of molecules is “5 orders of magnitude” greater than the number of locusts. That means it is 105 = 100,000 times greater. Comparing numbers by order of magnitude is a kind of ratio comparison because we are telling about how many times greater one number is than another.

Example: Write the numbers in scientific notation and compare the orders of magnitude.

The volume of the earth is about 1,100,000,000,000,000,000,000 m3 and the volume of the sun is about 1,400,000,000,000,000,000,000,000,000 m3.

volume of the earth, cubic meters 1.1 x 1021 volume of the sun, cubic meters 1.4 x 1027

The volume of the sun is 6 orders of magnitude greater than that of the earth. The sun is 106 = 1,000,000 = one million times larger than the earth.

Another reason scientists use scientific notation is that in many cases it indicates the accuracy of the measurement or calculation. Conventionally scientists use significant figures in the decimal part and the power of ten to indicate the magnitude.

Using scientific notation, you could indicate that the mountain measured 29,000 feet exactly by including the zeros in the decimal part: 2.9000 x 104. (By the way, in 1999 geographers using GPS technology measured the height of Mt. Everest to be 29,035 feet.)

!



Lesson 36: Scientific Notation

Worksheet Name __________________________________________

1. Write the power of 10 above each place value. The first few are done for you.

1. Write the number one hundred trillion in standard notation and as a power of ten.

2. Write the number 1011 in standard notation and in words.

3. Change scientific notation to standard notation.

a. 9.6 x 1020

b. 4.006621 x 1016

c. 7.15 x 108

4. Change standard notation to scientific notation.

a. 87,909,000,000,000,000

b. 807,000,000,000,000

c. 1,709,110,000,000,000,000,000

5. Write the underlined number in standard notation.

The real estate listings show the asking price is $1.8 million.

The company showed profits of $46.8 billion in 2010.

6. Write a place value name with a decimal point to convey the underlined numeric information.

Total costs of $40,500,000,000 for clean up.

About 4,900,000 barrels of oil.

7. Write the numbers in scientific notation.

At that time, the moon was about 399,000 km from earth, and Mars was 356 million km from earth.

8. Compare the orders of magnitude of the distances in #7.

About how many times greater is the distance to Mars than the distance to the moon?


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1

7, 3 6 9, 1 4 8, 0 2 5


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1

5 0 0 0 0 0 0 5 3 0 0 0 0


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1

3, 4 8 5, 2 1 8

109 108 107 106 105 104 103 102 101


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1

4 0 9, 0 0 0, 0 0 0


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1

7, 3 6 9, 1 4 8, 0 2 5


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1

5 0 0 0 0 0 0 5 3 0 0 0 0


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1

3, 4 8 5, 2 1 8

109 108 107 106 105 104 103 102 101


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1


100 10 1 100 10 1 100 10 1 100 10 1 100 10 1

4 0 9, 0 0 0, 0 0 0




Homework 36A Name __________________________________________

1. Use the formula C = (F – 32) / 1.8 to find the Celsius temperature equivalent to 98.6ºF.

What is special about 98.6ºF?

2. Find the height of an object shot into the air at 180 ft/sec when t = 11.25 seconds using the equation

h = !16t2+ 180t

What is special about the time t = 11.25?

3. Mark the number line with the following values: 0.5, 0.05, 0.49, 0.51.

4. Fill in the missing values in the chart.

5. Use the chart and the inch ruler to answer the questions.

a. Between which two decimals in the chart in #4 would 0.2 be located?

b. Between which two fractions on the ruler would 0.2 inches be located?

c. Between which two fractions on the ruler would 0.3 inches be located?

Fraction in Lowest Terms

Sixteenths Decimal Equivalent

1/16

2/16

3/16

4/16

5/16

6/16

7/16

8/16



1/16 1/16 0.0625

1/8 2/16 0.1250

3/16 3/16 0.1875

1/4 4/16 0.2500

5/16 5/16 0.3125

3/8 6/16 0.3750

7/16 7/16 0.4375

1/2 8/16 0.5000



6. Write a sentence to compare the data.

7. What is the ratio of Huff’s At Bats to Fontenot’s?

Write a sentence comparing the number of At Bats.

8. If you divide Hits by At Bats, then round to three decimal places, you have calculated the rate called the batting average. Compute the batting average for each player.

Write a sentence comparing the player’s batting averages.


a. 888,800,000,000

b. 5,009,000,000,000,000,000

c. 11,780,000,000

10. Write the numbers in standard notation.

a. 9.05 x 108

b. 7.992 x 1012

c. 5.0 x 107

11. a. Re-write with a place value name.

Human stone age cultures arose about 2,500,000 years ago.

b. Translate the written name to standard notation.

Evidence from radiometric dating indicates that the Earth is about 4.570 billion years old.

12. Write the numbers from #11 in scientific notation:

a. Human stone age culture:

b. Age of Earth:

c. What is the difference in order of magnitude of the two numbers?

d. How many times greater is the age of the earth than the time since stone age culture began?

13. Simplify, then solve the equation.

0.04 10,000 ! x( ) + 0.07x = 475

2010 Giants Season Statistics

Player At Bats Hits

Huff 569 165

Fontenot 240 68




Homework 36A Answers

1. Use the formula C = (F – 32) / 1.8 to find the Celsius temperature equivalent to 98.6ºF.

C = 98.6 ! 32( ) / 1.8

= 37ºC

What is special about 98.6ºF?

It is normal human body temperature.


h = !16t2+ 180t

h = !16 11.25( )2

+ 180 11.25( )= 0 feet


It is the exact time the object hits the ground.





0.1875 < 0.2 < 0.25


Between 3/16 and 1/4.


0.25 < 0.3 < 0.3125

Between 1/4 and 5/16



1/16

2/16

3/16

4/16

5/16

6/16

7/16

8/16



1/16 1/16 0.0625

1/8 2/16 0.1250

3/16 3/16 0.1875

1/4 4/16 0.2500

5/16 5/16 0.3125

3/8 6/16 0.3750

7/16 7/16 0.4375

1/2 8/16 0.5000




Huff had more at bats and more hits than teammate Fontenot.

7. What is the ratio of Huff’s At Bats to Fontenot’s?

569 / 240 = 2.37


Huff had more than twice as many at bats as Fontenot.


Huff: 165 / 569 = 0.290

Fontenot: 68 / 240 = 0.283


Huff has the higher batting average at 0.290, with Fontenot close at 0.283.


a. 888,800,000,000 8.888 x 1011

b. 5,009,000,000,000,000,000 5.009 x 1018

c. 11,780,000,000 1.178 x 1010


a. 9.05 x 108 905,000,000

b. 7.992 x 1012 7,992,000,000,000

c. 5.0 x 107 50,000,000


Human stone age cultures arose about 2,500,000 years ago.

2.5 million


Evidence from radiometric dating indicates that the Earth is about 4.570 billion years old.

4,570,000,000


a. Human stone age culture: 2,500,000 = 2.5 x 106

b. Age of Earth: 4.570 x 109

c. What is the difference in order of magnitude of the two numbers? The exponents are 9 and 6, so the difference is 9 – 6 = 3.

d. How many times greater is the age of the earth than the time since stone age culture began? 103 = 1000 times


0.04 10,000 ! x( ) + 0.07x = 475

400 ! 0.04x + 0.07x = 475

400 + (!0.04x + 0.07x) = 475

0.03x + 400 = 475

0.03x + 400 ! 400 = 475 ! 400

0.03x = 75 0.03x / 0.03 = 75 / 0.03

x = 2500

2010 Giants Season Statistics

Player At Bats Hits

Huff 569 165

Fontenot 240 68




Homework 36B Name _________________________________________

1. Use the formula C = (F – 32) / 1.8 to find the Celsius temperature equivalent to 212ºF.

What is special about 212ºF?


h = !16t2+ 252t










9/16

10/16

11/16

12/16

13/16

14/16

15/16

16/16



9/16 9/16 0.5625

5/8 10/16 0.6250

11/16 11/16 0.6875

3/4 12/16 0.7500

13/16 13/16 0.8125

7/8 14/16 0.8750

15/16 15/16 0.9375

1 16/16 1.0000




7. Find the difference of Suzuki’s At Bats to Tulowitski’s.





a. 204,800,000

b. 50,000,000,000,000,000,000

c. 1,780,000,000,000


a. 3.5 x 1018

b. 2.02 x 106

c. 4.111 x 1015


The total surface area of earth’s oceans is about 139,800,000 square miles.


The surface of the great Salt Lake is 1.8 thousand square miles.


a. Ocean Surface:

b. Salt Lake Surface:

c. What is the difference in order of magnitude of the two numbers?

d. How many times greater is the surface of the oceans than the surface of Salt Lake?


0.05 20,000 ! x( ) + 0.06x = 1070

2010 Major League Statistics

Player At Bats Hits

Suzuki (SEA) 680 214

Tulowitzki (COL) 470 148

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Lesson 36 Text - Diablo Valley Collegevoyager.dvc.edu/~lmonth/PreAlg/lesson36student.pdf · scientific notation for the result 670,800,000,000,000. Scientific Notation to Standard