Significant FiguresSignificant Figures

Integrated ScienceIntegrated Science

Dr. MayDr. May

Significant FiguresSignificant Figures

Numbers obtained from measurements are Numbers obtained from measurements are never exact valuesnever exact values

Maximum precision includes all digits that Maximum precision includes all digits that are known plus one estimatedare known plus one estimated

The digits used to express a measured The digits used to express a measured quantity are known as quantity are known as significant figuressignificant figures

Evaluating ZeroEvaluating Zero

In any measurement all nonzero numbers In any measurement all nonzero numbers are significantare significant

65.6291 grams has six significant figures65.6291 grams has six significant figures

Zeros may or may not be significant Zeros may or may not be significant depending on their position in the numberdepending on their position in the number

Zero is Significant WhenZero is Significant When

It is between nonzero digitsIt is between nonzero digits

2.05 has three significant figures2.05 has three significant figures

61.009 has five significant figures61.009 has five significant figures

Zero is Significant WhenZero is Significant When

It is at the end of a number that includes a It is at the end of a number that includes a decimal pointdecimal point

0.500 has three significant figures0.500 has three significant figures 25.160 has five significant figures25.160 has five significant figures 200. has three significant figures200. has three significant figures

Zero is Zero is NotNot Significant When Significant When

It comes before the first nonzero digit It comes before the first nonzero digit (These zeros are used to place the decimal)(These zeros are used to place the decimal)

0.000.002525 has two significant figures has two significant figures 0.000.00708708 has three significant figures has three significant figures

Zero Is Zero Is NotNot Significant When Significant When

It comes at the end of a number that It comes at the end of a number that contains no decimal pointcontains no decimal point

11000 has one significant figure000 has one significant figure 59590 has two significant figures0 has two significant figures

Determine Significant FiguresDetermine Significant Figures

4.5 inches4.5 inches == 3.025 feet3.025 feet == 125.0 meters125.0 meters == 0.001 miles0.001 miles == 25.0 grams25.0 grams == 100,000 people100,000 people == 205 birds205 birds ==

22

44

44

11

33

11

33

Rounding Off NumbersRounding Off Numbers

Integrated ScienceIntegrated Science

Dr. MayDr. May

Rounding Off NumbersRounding Off Numbers

When we do calculations we often obtain When we do calculations we often obtain answers with more digits than are justifiedanswers with more digits than are justified

We need to drop the excess digits to express We need to drop the excess digits to express the answer in the proper number of the answer in the proper number of significant figuressignificant figures

This is called This is called rounding off numbersrounding off numbers

Rounding Off Numbers - Rule 1Rounding Off Numbers - Rule 1

When the first digit after those you want to When the first digit after those you want to retain is 4 or less, that digit and all others to retain is 4 or less, that digit and all others to the right are dropped.the right are dropped.

The last digit retained is not changedThe last digit retained is not changed

Round 1.00629 to 4 significant figuresRound 1.00629 to 4 significant figures 1.0061.0062929 = 1.006 = 1.006

Rounding Off Numbers - Rule 2Rounding Off Numbers - Rule 2

When the first digit after those you want to When the first digit after those you want to retain is 5 or greater, that digit and all others retain is 5 or greater, that digit and all others to the right are dropped.to the right are dropped.

The last digit retained is increased by 1The last digit retained is increased by 1

Round 18.02500 to four significant figuresRound 18.02500 to four significant figures 18.0218.02500500 = 18.03 = 18.03

Round Off As IndicatedRound Off As Indicated

42.2442.2466 (four) (four) == 88.0188.0155 (four) (four) == 0.08960.089655 (three) (three) == 0.0890.0896565 (two) (two) == 225.225.33 (three) (three) == 14.114.150 50 (three)(three) ==

42.2542.25

88.0288.02

0.08970.0897

0.0900.090

225225

14.214.2

Scientific NotationScientific Notation

Integrated ScienceIntegrated Science

Dr. MayDr. May

Scientific NotationScientific Notation

Very large and very small numbers can be Very large and very small numbers can be simplified and conveniently written using a simplified and conveniently written using a power of 10power of 10

4,500,000,000 (4.5 billion) can be written4,500,000,000 (4.5 billion) can be written

4.5 x 104.5 x 1099

Writing a number as a power ofWriting a number as a power of 10 is called 10 is called scientific notationscientific notation

Powers of TenPowers of Ten

101000 ==

101011 ==

101022 ==

101033 ==

101044 ==

101055 ==

101066 ==

11

1010

100100

1,0001,000

10,00010,000

100,000100,000

1,000,0001,000,000

Negative Powers of TenNegative Powers of Ten

10 10 00 ==

10 10 11 ==

10 10 22 ==

10 10 33 ==

10 10 44 ==

10 10 55 ==

10 10 66 ==

11

0.10.1

0.010.01

0.0010.001

0.00010.0001

0.000010.00001

0.0000010.000001

Number to Scientific NotationNumber to Scientific Notation

Convert 0.000056 to 5.6 x 10 Convert 0.000056 to 5.6 x 10 55

Choose the number between 1 and 10 = 5.6Choose the number between 1 and 10 = 5.6 Multiply by 10: 5.6 x 10Multiply by 10: 5.6 x 10 If the number is < 1 use a negative exponentIf the number is < 1 use a negative exponent

5.6 x 10 5.6 x 10

Count the spaces the decimal was movedCount the spaces the decimal was moved

5.6 x 10 5.6 x 10 55

Number to Scientific NotationNumber to Scientific Notation

Convert 560,000 to 5.6 x 10 Convert 560,000 to 5.6 x 10 55

Choose the number between 1 and 10 = 5.6Choose the number between 1 and 10 = 5.6 Multiply by 10: 5.6 x 10Multiply by 10: 5.6 x 10 If the number is > 1 use a positive exponentIf the number is > 1 use a positive exponent

5.6 x 10 5.6 x 10 Count the spaces the decimal was movedCount the spaces the decimal was moved

5.6 x 10 5.6 x 10 55

Scientific Notation to NumberScientific Notation to Number

Convert 5.6 x 10 Convert 5.6 x 10 55 to 560,000 to 560,000 Write the significant figures = 56Write the significant figures = 56 The exponent is positive, the number is > 1The exponent is positive, the number is > 1 Add zeros to place the decimal 5 spaces Add zeros to place the decimal 5 spaces

to the to the rightright

560,000560,000..

55

Scientific Notation to NumberScientific Notation to Number

Convert 5.6 x 10 Convert 5.6 x 10 55 to 0.000056 to 0.000056 Write the significant figures = 56Write the significant figures = 56 The exponent is negative, the number is < 1The exponent is negative, the number is < 1 Add zeros to place the decimal 5 spaces Add zeros to place the decimal 5 spaces

to the to the leftleft

0.0000560.000056

55

Convert To Scientific NotationConvert To Scientific Notation

0.000340.00034 == 0.001450.00145 == 0.00009850.0000985 == 0.0168560.016856 == 0.00039670.0003967 == 0.00000020.0000002 == 0.000400.00040 == 0.006000.00600 ==

3.4 x 10 3.4 x 10 44

1.45 x 10 1.45 x 10 33

9.85 x 10 9.85 x 10 55

1.6856 x 10 1.6856 x 10 22

3.967 x 10 3.967 x 10 44

2 x 10 2 x 10 77

4.0 x 10 4.0 x 10 44

6.00 x 10 6.00 x 10 33

Convert To Scientific NotationConvert To Scientific Notation

34003400 = = 36,000,00036,000,000 = = 367,800,000,000 =367,800,000,000 = 5858

== 6578965789 = = 1,000,000,0001,000,000,000

== 2,0002,000 = =

3.4 x 103.4 x 1033

3.6 x 103.6 x 1077

3.678 x 103.678 x 101111

5.8 x 105.8 x 1011

6.5789 x 106.5789 x 1044

1 x 101 x 1099

2 x 102 x 1033

Convert to Numerical ValuesConvert to Numerical Values

7.4 x 107.4 x 1033 = = 5.6 x 105.6 x 1055 == 6.674 x 106.674 x 101010 = = 5.1 x 105.1 x 1044 = = 6.5559 x 106.5559 x 1011 == 3.64186 x 103.64186 x 1044 = = 1 x 101 x 1033 = =

7,4007,400

560,000560,000

66,740,000,00066,740,000,000

51,00051,000

65.55965.559

36,418.636,418.6

1,0001,000

Convert to Numerical ValuesConvert to Numerical Values

7.4 x 107.4 x 1033 = = 5.6 x 105.6 x 1055 == 6.674 x 106.674 x 1088 = = 5.1 x 105.1 x 1044 = = 6.5559 x 106.5559 x 1011 == 3.641 x 103.641 x 1044 = = 1 x 101 x 1033 = =

0.00740.0074

0.0000560.000056

0.000000066740.00000006674

0.000510.00051

0.655590.65559

0.00036410.0003641

0.0010.001

The EndThe End

This presentation was created for the benefit This presentation was created for the benefit of our students by the Science Department of our students by the Science Department at at Howard High School of TechnologyHoward High School of Technology

Please send suggestions and comments to Please send suggestions and comments to rmay@nccvt.k12.de.usrmay@nccvt.k12.de.us