significant digits

SIGNIFICANT DIGITSSIGNIFICANT DIGITS

We’ll call them “S.D.’s” for shortWe’ll call them “S.D.’s” for short

1 1 3

Why Use Them??Why Use Them??

To show how precise our instruments areTo show how precise our instruments are

• Whatever you measure you must estimate 1 decimal place past what you measure

• Represent the uncertainty of a piece of equipment by using + or –

• Read what the ruler/graduated beaker says and estimate 1 number past the marking

The following types of digits are considered significant:The following types of digits are considered significant:

any non-zero number…for example, 562any non-zero number…for example, 562this number has 3 S.D.’sthis number has 3 S.D.’s

all zeros between 2 other S.D.’s…for all zeros between 2 other S.D.’s…for example, 5006example, 5006

this number has 4 S.D.’sthis number has 4 S.D.’s If there is a decimal, any zero that follows If there is a decimal, any zero that follows

the last nonzero number…for example the last nonzero number…for example 560.0560.0

this number has 4 S.D.’sthis number has 4 S.D.’s

The following types of digits

are not considered significant

The following types of digits

are not considered significant

If there is no decimal point, any zero If there is no decimal point, any zero following the last nonzero digit is not following the last nonzero digit is not significant…for example, 9050significant…for example, 9050

this number has 3 S.D.’sthis number has 3 S.D.’s all zeros appearing in front of the first all zeros appearing in front of the first

nonzero digit…for example, .0091nonzero digit…for example, .0091

this number has 2 S.D.’sthis number has 2 S.D.’s

Now we can do mathematical operations the RIGHT WAY!!Now we can do mathematical operations the RIGHT WAY!!

ADDITION AND SUBTRACTIONADDITION AND SUBTRACTION

Consider the following problem Consider the following problem

5.0 + 15.5 + 20.25 = ???5.0 + 15.5 + 20.25 = ??? After you read the problem, do the addition After you read the problem, do the addition

(or subtraction) like you normally would.(or subtraction) like you normally would. Write down the entire answer from your Write down the entire answer from your

calculatorcalculator

you should have gotten 40.75you should have gotten 40.75

ADDITION AND SUBTRACTIONADDITION AND SUBTRACTION

WE ARE NOT DONE!!!!WE ARE NOT DONE!!!! Decide which was the Decide which was the LEAST LEAST

PRECISEPRECISE number in the problem number in the problem

That would be the 5.0 OR theThat would be the 5.0 OR the

15.5…..both go to the tenths place15.5…..both go to the tenths place For +/-, the answer goes to the same place!!For +/-, the answer goes to the same place!! 40.75 must go to the tenths……it becomes 40.75 must go to the tenths……it becomes 40.840.8

Whew….DONE!!!Whew….DONE!!!

Not Exactly…..Not Exactly…..

MULTIPLICATION AND DIVISIONMULTIPLICATION AND DIVISION

Consider the following problem Consider the following problem

562 x .0095 = ???562 x .0095 = ??? After you read the problem, do the After you read the problem, do the

multiplication (or division) like you normally multiplication (or division) like you normally would.would.

Write down the entire answer from your Write down the entire answer from your calculatorcalculator

you should have gotten 5.339you should have gotten 5.339

MULTIPLICATION AND DIVISIONMULTIPLICATION AND DIVISION

WE ARE NOT DONE!!!!WE ARE NOT DONE!!!! Decide which number in the problem had the Decide which number in the problem had the

LEAST NUMBERLEAST NUMBER OF S.D.’s OF S.D.’s

That would be the .0095 -- it only has 2 That would be the .0095 -- it only has 2 For * or /, the answer has the same number of For * or /, the answer has the same number of

S.D.’s as the number in the problem with the S.D.’s as the number in the problem with the fewest S.D.’s!!fewest S.D.’s!!

5.339 can only have 2 S.D.’s…it becomes5.339 can only have 2 S.D.’s…it becomes 5.35.3

Rounding Off Nonsignificant DigitsRounding Off Nonsignificant Digits

If a calculator displays 12.846239 and only 3 If a calculator displays 12.846239 and only 3 significant digits are justified, use the rounding significant digits are justified, use the rounding rules…rules…

Beware of “Place-Holder Zeros” !!!Beware of “Place-Holder Zeros” !!!

Round off 151.1 to 2 sig digits?Round off 151.1 to 2 sig digits?

Round off 14652.83 to 3 sig digits?Round off 14652.83 to 3 sig digits?

Rounding Off Nonsignificant DigitsRounding Off Nonsignificant Digits

1.1. If the first nonsignificant digit is less that 5, drop If the first nonsignificant digit is less that 5, drop all nonsignificant digitall nonsignificant digit

2.2. If the first nonsignificant digit is If the first nonsignificant digit is ≤5, increase the ≤5, increase the last significant digit by 1 and drop all last significant digit by 1 and drop all nonsignificant digits.nonsignificant digits.

3.3. If a calculation has 2 or more operations, retain If a calculation has 2 or more operations, retain the nonsignificant digits until the final operation.the nonsignificant digits until the final operation.

ALL DONE!!!ALL DONE!!!

This has been….This has been….

A Baron & Baron ProductionA Baron & Baron Production

Special thanks to Mrs. Robin Special thanks to Mrs. Robin McNemarMcNemar