*
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
*
* 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 would be read
as 26.45g. 26.4 measured, .05 estimated)
The estimated digit
adds to the
accuracy of the
measurement
*
* 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 10ml
2) record the real measurement… 110ml
3) estimate the distance traveled
toward the next real measurement… half
way is ~5ml
4) record your measurement using the
real and estimated digits… 115ml
*
* Rule #1: Trailing zeros without a decimal are NEVER
significant (13000… only the 13 are significant)
* Rule #2: Leading zeros are NEVER significant (0.0027… only
the 27 are significant)
* Zeros that simply hold places are not significant
* Rule #3: Trailing zeros after a decimal are ALWAYS
significant (0.0024500… 24500 are all significant and add to
the precision of the measurement)
* Rule #4: Zeros found between numbers are ALWAYS
significant (987001… all numbers are significant)
*
* 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)
*
* 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)
*
* 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 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 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%
* 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%