*Working With Numbers

Objectives:

*1. Define significant digits.

*2. Explain how to determine which digits in measurement are significant.

*3. Convert measurements in to scientific notation.

Key Terms:

significant digit, percent error, density

*Significant Digits*The purpose of significant digits is to limit the amount

of uncertainty in the data you display. The measured and the estimated digits of a measurement are considered significant. (If you look at the picture the significant digits what would be the measurement?)

The estimated digit

adds to the

accuracy of the

measurement

*Significant Digits*If you look at the picture the significant digits would be

read as 26.45g.

*26.4 measured

*.05 estimated

The estimated digit

adds to the

accuracy of the

measurement

*Estimated Digits & Measurements

*When making measurements in science it is important to write down all of the digits that a device can give you and one estimated digit. Remember that measurements are never completely accurate for the following two reasons:*measuring equipment is never completely free of flaws

*measurements always involve some degree of estimation

Note:

*Electronic devices take care of estimation for you and record the last digit as the estimated digit.

When recording the measurement of a piece of scientific equipment try following the following steps: 1) determine the scale increments… ea. graduation is ___ml 2) record the real measurement… ___ml 3) estimate the distance traveled toward the next real measurement… half way is …___ml 4) What is the final measurement?

1) scale increments… ea. graduation is 10ml 2) real measurement… 110ml 3) estimate the distance traveled toward the next real measurement… half way is ~5ml 4) recorded measurement is 115ml

110ml real 5ml estimated

*Rules having to do with ZEROS!

*Zeros that simply hold places are not significant

*Rule #1: Trailing zeros without a decimal are NEVER significant (13000… only the 1 & 3 are significant)

*This number is said to have 2 significant digits

*Rule #2: Leading zeros are NEVER significant (0.0027… only the 2 & 7 are significant)

*Rule #3: Trailing zeros after a decimal are ALWAYS significant (0.0024500… 24500 are all significant and add to the accuracy of the measurement)

*Rule #4: Zeros found between numbers are ALWAYS significant (987001… all numbers are significant)

*This number is said to have 6 significant digits

*Significant Digits In Calculations –

Multiplication and Division*The measurement with the smallest amount of

significant digits determines the significant digits of the answer

Example:

0.3287g x 45.2g = ?

1) determine the operand with the smallest amount of significant digits (0.3287 (___ sig digits) & 45.2 (__ sig digits))

2) perform the operation on your calculator and round to the correct amount of digits

0.3287g x 45.2g = 14.85724g (too many digits)

____g (rounded to ___ sig digits)

*Significant Digits In Calculations –

Multiplication and Division*The measurement with the smallest amount of

significant digits determines the significant digits of the answer

Example:

0.3287g x 45.2g = ?

1) determine the operand with the smallest amount of significant digits (0.3287 (4 sig digits) & 45.2 (3 sig digits))

2) perform the operation on your calculator and round to the correct amount of digits

0.3287g x 45.2g = 14.85724g (too many digits)

14.9g (rounded to 3 sig digits)

*Significant Digits In Calculations – Addition

and Subtraction*The total cannot be more accurate than the least accurate

measurement. This time the amount of significant digits of each number does not matter. The quantity with the least digits to the right of the decimal point determines the accuracy of the answer

Example:

125.5kg + 52.68kg + 2.1kg = ?

1) determine the precision of your least accurate measurement. (125.5kg and 2.1kg are both accurate to the ___ place while 52.68kg is accurate to the ___ place)

2) perform the operation

125.5kg + 52.68kg + 2.1kg = 180.28kg (too precise!)

___kg (precise to the

___place)

*Significant Digits In Calculations – Addition

and Subtraction*The total cannot be more accurate than the least accurate

measurement. This time the amount of significant digits of each number does not matter. The quantity with the least digits to the right of the decimal point determines the accuracy of the answer

Example:

125.5kg + 52.68kg + 2.1kg = ?

1) determine the precision of your least accurate measurement. (125.5kg and 2.1kg are both accurate to the tenths place while 52.68kg is accurate to the hundredths place)

2) perform the operation

125.5kg + 52.68kg + 2.1kg = 180.28kg (too precise!)

180.3kg (precise to the

tenths place)

*Rules with Mixed Operations

*When doing combinations of addition/subtraction and multiplication/division, each step determines its significant digits. The digits in the end answer are determined by the last operation performed.

Example:

1250cal – (234.207cal/52.69cal) = ?

1250cal – (4.445cal) =

(parenthesis 1st … answer has 4 sig digits)

1250cal – (4.445cal) = 1245.555cal (too precise!)

1250cal (accurate to the tens

place)

*Scientific Notation*Scientific notation is a way that scientists make incredibly large

numbers used in science easier to work with. There are 602,000,000,000,000,000,000,000 atoms in a mole of a substance. It is much easier to use the answer as 6.02 x 1023

Rules:

*1) The answer must be in the form of a real number followed by a decimal point while retaining the correct amount of significant digits.

*2) If the magnitude of the number is to be reduced the exponent will be positive.

Example: 398700 = 3.987 x 105 (the exponent is equal to the

number of times that the decimal point was moved)

*3) If the magnitude of the number is to be increased the exponent will be negative.

Example: 0.00501 = 5.01 x 10-3

*Percent Error *Percent error calculations are used to compare test results

to a known accepted quantity. The formula is as follows:

Percent error = ((measured value - accepted value) / accepted value ) * 100%

Percent error = ((experimental value – theoretical value) / theoretical value ) * 100%

*Note: The result can be positive or negative but the answer is always represented as the absolute value

Example:

The accepted mass of an object is 5.00g. When you measure it on your digital scale the reading shows 5.02g. What is the percent error of your measurement?

(5.02g – 5.00g) / 5.00g = 0.004 or 0.4%