significant figures

Significant Significant FiguresFigures

What is so What is so significant about significant about

numbers??numbers??

Just which numbers are Just which numbers are significant?significant?

All nonzero numbers All nonzero numbers (all numbers 1-(all numbers 1-9) –9) –

289289 573573 119119


All nonzero numbers All nonzero numbers (all numbers 1-9) - (all numbers 1-9) - 289289

All zeros b/w numbers – 909All zeros b/w numbers – 909 501501 30023002 40000014000001


All nonzero numbers All nonzero numbers (all numbers 1-9) - (all numbers 1-9) - 289289

All zeros b/w numbers – 909All zeros b/w numbers – 909 Zeros to the right of a number Zeros to the right of a number

AND to the left of a written AND to the left of a written decimal point –decimal point – 250.250. 700.700. 10.10.


All nonzero numbers All nonzero numbers (all numbers 1-9) - 289(all numbers 1-9) - 289

All zeros b/w numbers – 909All zeros b/w numbers – 909 Zeros to the right of a number AND to the left of Zeros to the right of a number AND to the left of

a written decimal point – 250.a written decimal point – 250.700.700. 10.10. Zeros to the right of a number AND to Zeros to the right of a number AND to

the right or a written decimal point – the right or a written decimal point – 5.0 5.0 236.70236.70 3.0 x 103.0 x 1088

These are These are NEVERNEVER significantsignificant

Zeros to the left of the Zeros to the left of the decimal in numbers less decimal in numbers less than onethan one0.80.80.2220.222*** these are placeholders *** these are placeholders onlyonly

These are These are NEVERNEVER significantsignificant

Zeros to the left of the decimal Zeros to the left of the decimal in numbers less than onein numbers less than one

Zeros to the right of a Zeros to the right of a decimal, but to the left of decimal, but to the left of the first numberthe first number 0.0080.008 0.020.02

Exceptions to the Exceptions to the rulerule

Conversion factors – Conversion factors – unlimited sig. fig.unlimited sig. fig. 1 km/1000 m1 km/1000 m 60 min/1 hr60 min/1 hr 100 cm/1 m100 cm/1 m

Exceptions to the Exceptions to the rulerule

Conversion factors – unlimited Conversion factors – unlimited sig. fig.sig. fig.

Counting numbersCounting numbers 30 days30 days 12 in one dozen12 in one dozen

Calculations with sig. Calculations with sig. figs.figs.

The following rules must The following rules must be followed so that be followed so that results reflect sig. figs. results reflect sig. figs. from your measurements from your measurements taken originally…taken originally…

Addition & subtraction Addition & subtraction with sig. figs.with sig. figs.

Your answer can’t be MORE Your answer can’t be MORE precise than your LEAST precise than your LEAST precise measurement precise measurement

(what in the (what in the world does that mean??)world does that mean??)

Addition & subtraction Addition & subtraction with sig. figs.with sig. figs.

Your answer can’t be MORE Your answer can’t be MORE precise than your LEAST precise precise than your LEAST precise measurementmeasurement

Your answer must be rounded to Your answer must be rounded to the same number of decimal the same number of decimal places from the original problem…places from the original problem…

34.2 + 2.002 = ?34.2 + 2.002 = ?

Addition & subtractionAddition & subtraction

34.2 + 2.002 =34.2 + 2.002 =Actual answer would be Actual answer would be

36.20236.202


Actual answer would be 36.202Actual answer would be 36.202 The least precise number is The least precise number is

34.2, therefore your answer 34.2, therefore your answer MUST be to one decimal placeMUST be to one decimal place


Actual answer would be 36.202Actual answer would be 36.202 The least precise number is 34.2, The least precise number is 34.2,

therefore your answer MUST be to therefore your answer MUST be to one decimal placeone decimal place

Correct answer = 36.2Correct answer = 36.2


Example #2Example #23.999 – 1.77 = ?3.999 – 1.77 = ?


3.999 – 1.77 =3.999 – 1.77 =Actual answer would be Actual answer would be

2.2292.229



therefore your answer MUST be to therefore your answer MUST be to two decimal placestwo decimal places



therefore your answer MUST be to therefore your answer MUST be to two decimal placestwo decimal places

The correct answer would The correct answer would be 2.23be 2.23

Multiplication & divisionMultiplication & division

Answer must be in the Answer must be in the fewestfewest number of sig. figs. from the number of sig. figs. from the original problem.original problem.

Example:Example: 45.6 x 1.009 = ?45.6 x 1.009 = ?



Example:Example: 45.6 x 1.009 = ?45.6 x 1.009 = ? Actual is 46.0104Actual is 46.0104



Example:Example: 45.6 x 1.009 = ?45.6 x 1.009 = ? Actual is 46.0104Actual is 46.0104 Smallest # of sig figs is 3… so answer Smallest # of sig figs is 3… so answer

must be in 3 sig figs.must be in 3 sig figs.


Answer must be in the Answer must be in the fewestfewest number of sig. figs. from the original number of sig. figs. from the original problem.problem.

Example:Example: 45.6 x 1.009 = ?45.6 x 1.009 = ? Actual is 46.0104Actual is 46.0104 Smallest # of sig figs is 3… so answer Smallest # of sig figs is 3… so answer

must be in 3 sig figs.must be in 3 sig figs. Correct answer is 46.0 (not just 46)Correct answer is 46.0 (not just 46)


Example #2Example #2505 / 7 = ?505 / 7 = ?


Example #2Example #2 505 / 7 = ?505 / 7 = ? Actual answer is 72.14285714Actual answer is 72.14285714


Example #2Example #2 505 / 7 = ?505 / 7 = ? Actual answer is 72.14285714Actual answer is 72.14285714 Least number of sig figs is 1, so Least number of sig figs is 1, so

answer can only have one sig fig.answer can only have one sig fig.


Example #2Example #2 505 / 7 = ?505 / 7 = ? Actual answer is 72.14285714Actual answer is 72.14285714 Least number of sig figs is 1, so Least number of sig figs is 1, so

answer can only have on sig fig.answer can only have on sig fig. Correct answer is 70 or it Correct answer is 70 or it

could be 7 x 10could be 7 x 1011

Rounding…Rounding…

If the number right past the one If the number right past the one you want to keep is:you want to keep is: Greater than 5 Greater than 5 go up by one go up by one

295.46 to 4 sig figs would be 295.5295.46 to 4 sig figs would be 295.5 999.97 to four sig figs would be 999.97 to four sig figs would be

1000.1000.


If the number right past the one If the number right past the one you want to keep is:you want to keep is: Greater than 5 Greater than 5 go up by one go up by one Less than 5 Less than 5 no change no change

999.94 to four sig figs would be 999.94 to four sig figs would be 999.9999.9

564.44 to three sig figs would be 564564.44 to three sig figs would be 564


If the number right past the one If the number right past the one you want to keep is:you want to keep is: Greater than 5 Greater than 5 go up by one go up by one Less than 5 Less than 5 no change no change 5 followed by a number 5 followed by a number go up go up

by oneby one 2.352 to two sig figs would be 2.42.352 to two sig figs would be 2.4 4.156 to two sig figs would be 4.24.156 to two sig figs would be 4.2


If the number right past the one you If the number right past the one you want to keep is:want to keep is: Greater than 5 Greater than 5 go up by one go up by one Less than 5 Less than 5 no change no change 5 followed by a number 5 followed by a number go up by one go up by one 5 followed by nothing… look at 5 followed by nothing… look at

the number before it… if it is ODD the number before it… if it is ODD go up by one go up by one

3.375 to three sig figs would be 3.383.375 to three sig figs would be 3.38 0.035 to one sig fig would be 0.040.035 to one sig fig would be 0.04

Rounding…Rounding… If the number right past the one you want to If the number right past the one you want to

keep is:keep is: Greater than 5 Greater than 5 go up by one go up by one Less than 5 Less than 5 no change no change 5 followed by a number 5 followed by a number go up by one go up by one 5 followed by nothing… look at the number before it… if it 5 followed by nothing… look at the number before it… if it

is ODD is ODD go up by one go up by one

5 followed by nothing… look at the 5 followed by nothing… look at the number before it… if it is EVEN number before it… if it is EVEN no no changechange

4.8785 to four sig figs would be 4.8784.8785 to four sig figs would be 4.878 399.345 to five sig figs would be 399.34399.345 to five sig figs would be 399.34

Scientific notationScientific notation

Very large or very small numbers Very large or very small numbers are expressed in scientific notationare expressed in scientific notation



M x 10M x 10n n



M x 10M x 10nn

““M” must be 1 or greater, but less M” must be 1 or greater, but less than 10than 10 2.2 x 102.2 x 1055


Very large or very small numbers are Very large or very small numbers are expressed in scientific notationexpressed in scientific notation

M x 10M x 10nn

““M” must be 1 or greater, but less than M” must be 1 or greater, but less than 1010

All numbers that represent “M” are All numbers that represent “M” are significantsignificant 3.10 x 103.10 x 1099

9.98 x 109.98 x 1033

Scientific notationScientific notation Very large or very small numbers are Very large or very small numbers are

expressed in scientific notationexpressed in scientific notation M x 10M x 10nn

““M” must be 1 or greater, but less than 10M” must be 1 or greater, but less than 10 All numbers that represent “M” are All numbers that represent “M” are

significantsignificant Numbers that represent “n” are whole Numbers that represent “n” are whole

numbers, positive or negativenumbers, positive or negative 6.022 x 106.022 x 102323

2.1 x 102.1 x 10-5-5

Scientific notationScientific notation Very large or very small numbers are expressed in scientific Very large or very small numbers are expressed in scientific

notationnotation M x 10M x 10nn

““M” must be 1 or greater, but less than 10M” must be 1 or greater, but less than 10 All numbers that represent “M” are significantAll numbers that represent “M” are significant Numbers that represent “n” are whole numbers, positive or Numbers that represent “n” are whole numbers, positive or

negativenegative

Examples:Examples: 270,000 would be 2.7 x 10270,000 would be 2.7 x 1055

The understood decimal is behind the last zero…The understood decimal is behind the last zero…move the decimal until it makes the number move the decimal until it makes the number represent “M”represent “M”

0.000000505 would be 5.05 x 100.000000505 would be 5.05 x 10-7-7

Calculations w/ scientific Calculations w/ scientific notation notation

Addition and subtraction:Addition and subtraction: All numbers must have the same exponent All numbers must have the same exponent

before “M” can be added or subtractedbefore “M” can be added or subtracted Answer will have the same exponent value as Answer will have the same exponent value as

originalsoriginals

Don’t forget rules w/ sig figs… they Don’t forget rules w/ sig figs… they still apply!!still apply!! 2.4 x 102.4 x 1022 + 5.7 x 10 + 5.7 x 1022 = 8.1 x 10 = 8.1 x 1022

9.05 x 109.05 x 1055 – 5.5 x 10 – 5.5 x 1055 = actual 3.55 x = actual 3.55 x 101055

Correct is 3.6 x 10Correct is 3.6 x 1055


Addition and subtractionAddition and subtraction:: All numbers must have the same All numbers must have the same

exponent before “M” can be added or exponent before “M” can be added or subtractedsubtracted

The problem comes when exponents are The problem comes when exponents are DIFFERENT… DIFFERENT…

You must MAKE the exponents the same…You must MAKE the exponents the same… LL / RR = left larger or right reduce LL / RR = left larger or right reduce

when you move the decimal it changes the when you move the decimal it changes the exponent -- ALWAYSexponent -- ALWAYS


Addition and subtractionAddition and subtraction:: 2.5 x 102.5 x 1044 + 5.2 x 10 + 5.2 x 1033 = = Choose which number you want to deal Choose which number you want to deal

with… either will work outwith… either will work out



with… either will work outwith… either will work out I like to change the smaller exponent to the I like to change the smaller exponent to the

larger larger 5.2 x 10 5.2 x 1033




larger larger 5.2 x 10 5.2 x 1033 I want to make the exp. I want to make the exp. largerlarger, so I move , so I move

the decimal one place to the the decimal one place to the leftleft





the decimal one place to the the decimal one place to the leftleft This makes it This makes it .52 x 10.52 x 1044


Addition and subtraction:Addition and subtraction: 2.5 x 102.5 x 1044 + 5.2 x 10 + 5.2 x 1033 = = Choose which number you want to deal Choose which number you want to deal



the decimal one place to the the decimal one place to the leftleft This makes it This makes it .52 x 10.52 x 1044

Now “M” can be added because exp are the Now “M” can be added because exp are the samesame

Actual answer = 3.02 x 10Actual answer = 3.02 x 1044

Correct answer = 3.0 x 10Correct answer = 3.0 x 1044

Calculations w/ scientific Calculations w/ scientific notationnotation

Multiplication – Multiplication – this one is much easier this one is much easier 3.0 x 103.0 x 1033 x 4.0 x 10 x 4.0 x 1044



You will multiply your “M”s You will multiply your “M”s together (12)together (12)



You will multiply your “M”s together You will multiply your “M”s together (12)(12)

Then add your “n”s together (7)Then add your “n”s together (7)



You will multiply your “M”s together You will multiply your “M”s together (12)(12)

Then add your “n”s together (7)Then add your “n”s together (7) CAREFUL… you will often need to CAREFUL… you will often need to

move the decimal to keep correct move the decimal to keep correct scientific notationscientific notation



You will multiply your “M”s together (12)You will multiply your “M”s together (12) Then add your “n”s together (7)Then add your “n”s together (7) CAREFUL… you will often need to move CAREFUL… you will often need to move

the decimal to keep correct scientific the decimal to keep correct scientific notationnotation

Actual answer would be 12 x 10Actual answer would be 12 x 1077



You will multiply your “M”s together (12)You will multiply your “M”s together (12) Then add your “n”s together (7)Then add your “n”s together (7) CAREFUL… you will often need to move the CAREFUL… you will often need to move the

decimal to keep correct scientific notationdecimal to keep correct scientific notation Actual answer would be 12 x 10Actual answer would be 12 x 1077

Move the decimal one place to the Move the decimal one place to the LEFT to make 1.2LEFT to make 1.2





Move the decimal one place to the LEFT to Move the decimal one place to the LEFT to make 1.2make 1.2

This makes the exponent go up (larger) This makes the exponent go up (larger) to 8to 8





Move the decimal one place to the LEFT to make Move the decimal one place to the LEFT to make 1.21.2

This makes the exponent go up (larger) to 8This makes the exponent go up (larger) to 8 Now you answer is 1.2 x 10Now you answer is 1.2 x 108 8 (2 sig figs)(2 sig figs)

More examples…More examples…

significant figures

Documents

nonzero numbers

decimal placecorrect

precise number

zeros bw numbers

precise measurementyour

number of decimal places

fewest number of

written decimal point