MATH SKILLS IN PHYSICS

Significant Figures
Two kinds of numbers
i. exact numbers: example: There are exactly 20 apples in a box
                                              Most cats have exactly 2 ears and 4 legs. 

ii.inexact numbers: example: any measurement
the width of a piece of  paper, 220 mm     (2 significant figures). 
                        more precise, 216 mm       (3 significant figures). 
                        more precise, 215.6 mm    (4 significant figures). 

PRECISION VERSUS ACCURACY

Accuracy refers to how closely a measured value agrees with the correct value.
Precision refers to how closely individual measurements agree with each other.

The number of significant figures is the number of digits believed to be correct by the person doing the measuring. It includes one estimated digit.  

Rules for Working with Significant Figures:

1. Leading zeros are never significant.
   Imbedded zeros are always significant.
   Trailing zeros are significant only if the decimal point is specified.
   Hint: Change the number to scientific notation. It is easier to see.
2. Addition or Subtraction:
   The last digit retained is set by the first doubtful digit.
3. Multiplication or Division:
   The answer contains no more significant figures than the least accurately known number.

Examples:

Going from left to right, the first non-zero digit is the first significant figure.

Each digit to its right is a significant figure.

To write a number x to n significant figures means to write down the number with n significant figures which is closest to x.

For example,  consider 0.000026857
To 1 significant figure it is 0.00003

To 2 significant figures it is 0.000027

To 3 significant figures it is 0.0000279
To 4 significant figures it is 0.00002686

Consider  4.198.

To 1 significant figure it is 4

To 2 significant figures it is 4.2

To 3 significant figures it is 4.20
(we write 4.20 instead of 4.2 to show that we have 3 significant figures).

To 4 significant figures it is 4.198

Consider 58 651

To 1 significant figure it is 60 000

To 2 significant figure it is 59 000

To 3 significant figure it is 58 700

To 4 significant figure it is 58 650

To 5 significant figure it is 58 651

Attentively, significant figures for whole numbers can be done using scientific notation.

Consider the number 70 000.

To 1 significant figure it is 7 x 10⁴

To 2 significant figures it is 7.0 x 10⁴

To 3 significant figures it is 7.00 x 10⁴

To 4 significant figures it is 7.000 x 10⁴

Example	Number of Significant Figures	Scientific Notation
0.00682	3	6.82 x 10-3
1.072	4	1.072 (x 100)
300	1	3 x 102
300.	3	3.00 x 102
300.0	4	3.000 x 102

 ^{Addition :  4.7832} 

                   ^1.234 

                ^{+ 2.02}

                   ^{8.0372  > Rounding  8.04}

 ^{> The answer must be rounded off to 3 significant figures, since 2.02 only has 3 significant figures.}

^{Subtraction : 1.0236}

                   ^{- 0.97268} 

                     ^{0.05092 > Rounding 0.0509}

>Subtraction is interesting when concerned with significant figures. Even though both numbers involved in the subtraction have 5 significant figures, the answer only has 3 significant figures when rounded correctly. Remember, the answer must only have 1 doubtful digit

Multiplication : 2.8723 x 1.6 = 4.59568 > Rounding 4.6

Division :  45.2   = 7.1093775 > Rounding 7.11

                                 ^{6. 3578}

Notes on Rounding

• When rounding off numbers to a certain number of significant figures, do so to the nearest value. 
• example: Round to 3 significant figures: 2.3467 x 10⁴ 
• (Answer: 2.35 x 10⁴)
• example: Round to 2 significant figures: 1.612 x 10³ 
•  (Answer: 1.6 x 10³)
• What happens if there is a 5? There is an arbitrary rule:
• If the number before the 5 is odd, round up.
• If the number before the 5 is even, let it be.
  The justification for this is that in the course of a series of many calculations, any rounding errors will be averaged out.
• example: Round to 2 significant figures: 2.35 x 10² 
• (Answer: 2.4 x 10²)
• example: Round to 2 significant figures: 2.45 x 10² 
• (Answer: 2.4 x 10²)
• Of course, if we round to 2 significant figures: 2.451 x 10², the answer is definitely 2.5 x 10² since 2.451 x 10² is closer to 2.5 x 10² than 2.4 x 10².