Key concepts for Chapter 2
2.1
All measurements have uncertainties. Measurements are never exact.
2.2
The uncertainty in measurement is reflected / indicated by using significant figures (or significant digits).
- the digits (s.f.) for a given measurement tells the reader that the last digit (right-most) is the
digit where there is uncertainty and that all other digits are certain.
- The number of significant figures for a number is the total number of digits the number has,
counting from left to right.
o EXCEPT in the follow 2 cases.
 If a number is less than 1 (i.e. 0.123, 0.0555 etc) but greater than 0, all zeroes
before (to the left of) the first non-zero digit are not significant.
 ex. 0.0023 has 2 s.f.
0.1507 has 4 s.f.
 If a number is a whole number and ends with 0s but without a decimal point
shown, all zeroes to the right of the last (right most) non-zero digit are not
significant.
 Ex. 7000 has 1 s.f.
20700 has 3 s.f.
 The same two rules apply when the numbers are negative.
- exact numbers have infinite number of significant figures
o exact numbers are those that are counted and not measured(2 eggs, 3 years, 5 cars)
o exact numbers are those that are definitions (i.e. 1 meter = 1000 millimeters)
2.3
Rounding of significant figures is the same as normal mathematical rounding.
- if the 1st non-significant digit is less than 5, drop all non s.f. digits
- if the 1st non-significant digit is 5 or more, drop all non s.f. digits and increase the last
significant digit by 1
- if a calculation involves multiple steps, wait until all calculations are done before rounding to
minimize error due to rounding.
2.4
Adding and Subtracting significant figures
- perform the calculation normal,
- examine each number and determine where the uncertain digits are for each of the number.
o Determine the unit/digit (i.e. 10s, 1s, tenths, hundredths, etc) with the highest
uncertainty (left most)
- transfer that digit down to answer
o so if the digit with the highest uncertainty is in the 10s digit, then the uncertainty in
the answer will be in the 10s digit
- round off non-significant digits in the answer
2.5
Multiplying and dividing significant figures
- perform calculations normally
- examine each number for the number of s.f. each number has
o determine what is the least number of s.f. all the numbers have
- the least number of s.f. is the number of s.f. the answer will have
- round off non-significant digits in the answer
2.6
Exponential numbers
- are numbers with a superscript
o which is defined as the number multiplied by itself a number of times equal to the
superscript.
 So, 63 = 6x6x6
102 = 10x10
24 = 2x2x2x2
- know the conversion of powers of 10s.
2.7
Scientific Notation
- a method used to standardize writing of numbers, especially useful when writing really large
or small numbers
- always has the format of;
 #.### x 10n
 Start with a non-zero digit, followed by a decimal point, followed by the other
significant digit(s)
 i.e.. 6.23
5.11
2
 (exception, if there is only 1 s.f., decimal point is not needed)
 That number is then multiplied by the correct powers of 10.
 i.e. 105
10-7
102
0
 (exception, 10 , which equals 1, is not used)
2.8
Unit equation and unit factors
- unit equation is just an equation that relates two equivalent quantities
o example:
1 dollar = 100 pennies
- unit factor (also known as conversion factor) is simple two fractions that can be written using
the two quantities
o example: quantities A & B -> A/B and
B/A
2.9
Unit analysis problem solving (ex. How many days in 2.5 years)
- write down the unit asked for in the answer
o
=
days
- write down the units related to the answer that are given
o
2.5 years
- Find the unit equation relating the units
o Years and days  1 year = 365 days
- Write the conversion factors
o 1 year / 365 days
365 days / 1 year
- Plug in the conversion factor into the equation and find that will give the right units in the
answer.
o 2.5 years x (1 year/365 days) = 2.5 yr2 / 365 days
WRONG units
o 2.5 years x (365 days / 1 year) = 912.5 days
CORRECT units
- round off non significant digits in answer
o 912.5 days  912 days
(2.5 years is an exact number, it’s not measured)
2.10
Percentages (%)
- are defined as the number parts (of a given quantity or quantities) per 100 parts of the sum
(total sample).
- Mathematically it is:
% = (part(s) / sum ) x 100
o Example: 2 apples and 8 oranges in a basket, what percentage do the apples make up
 (Part / sum ) * 100 = %
 [2 apples / (2 apples + 8 oranges)] * 100 = (2 / 10) *100 = 0.2*100 = 20%