Significant Figures & Precision

Learning Objective: to understand the meaning of significant figures, to be able to calculate numbers to the correct significance, and to understand how significant figures relate to the precision of a measurement.

Precision: I asked my students how old they were and their answers ranged from 15 to 16. One student said 15 and a half, another said “I’m almost 16”. Much to my surprise, no one answered “I’m 15 years, 2 months, 4 days, 17 hours, 18 minutes and 43 – no wait – 44 seconds old”!  The question to ask is how large or small a quantity do you need to properly represent something. In most conversations, saying you are 15 or 16 is good enough. We could say that you approximate your age to the last full year.  So the degree of precision is 1 year. You’re either 15 or 16 – that’s it. To be more precise, you might say that you are 15 and 1/2. Then the degree of precision would be 1/2 year. You can keep getting more and more precise to months, days, hours and seconds or even to 1/1000ths of a second.  This would be a very high degree of precision to be sure.

How does precision affect mathematical calculations?

The least precise measurement or number determines the overall precision of a calculation.

Consider the number 3. Is 3 really the same as 3.0?  The answer is that you don’t have enough information to tell for sure.  In fact the number 3 actually represents any number from 2.5 to 3.4999… which has been rounded to the nearest integer. 2.5 rounds up to 3 as does 2.6, 2.7, 2.8, and 2.9.  3.1 rounds down to 3 as does 3.2, 3.3 etc. right up through 3.4.

Now for a more difficult problem.  Is 3×3 the same as 9?

The smallest possible value for 3 could be 2.5.                                      2.5×2.5 = 6.25 rounds down to 6.                                                                       The maximum value of 3 could be 3.4999.                                                             3.4999 x3.4999  =  12.                                                                                             Take the average of these results (6 + 12) / 2 = 18/2 =9.

 

 

 

Orders of Magnitude aka Powers of 10

Learning objective: To understand powers of 10,  know how to multiply and divide powers of 10, and know how to convert from decimal to scientific and engineering notation.

Part 1: What are Orders of Magnitude and powers of 10?
So you’re sitting at coffee shop and some Physicists are writing order of magnitude calculations on the back of an envelope. You want to listen in and comment on the discussion but first you need to understand what they’re talking about.  This post is a simple study guide to help you understand and use powers of 10 and orders of magnitude.

First some simple definitions;

Decimal notation:  3.14159 ie numbers and a decimal point. More examples: 73,015.238,  1.059,  0.00063 etc.

Scientific notation: a decimal number between 1.0 and 9.999… multiplied by 10 raised to some power (also called an exponent).     scientific notation examples: 6.022×10^23 (Avogadro’s number), 1×10^100 a 1 followed by 100 zeros also known as 1 google, 1.602×10^-19 the charge of an electron, 6.67×10-11 the gravitational constant………. and the list goes on. Notice that the carrot “^” in 10^23 means 10 is raised to the 23rd power. This is the same as saying 1 followed by 23 zeros.  Formally, the correct way to write a number in scientific notation is 10 with the exponent written as a superscript but this is less common today because it’s easier to type ^ than to make a superscript.

Exponential notation: same as scientific notation

Engineering notation: Engineers started typing equations a long time ago, there was no “^” key on most of the old typewriters so engineers simply used the letter “e” or “E” to mean exponential notation.  For example, 6.022×10^23 could be written as 6.022e23 or as 6.022E23.  Negative exponents work the same way so 1.062×10^-19 = 1.062e-19 = 1.062E-19.

Order of magnitude: the exponent only, note round the number part down to 1 or up to 10 and adjust the exponent.                    examples: 1.062×10^-19, 1.062 rounds down to 1×10^-19 so the order of magnitude is -19 while 6.022×10^23 rounds up to 10×10^23 = 1×10^24 has an order of magnitude of 24.

Simple! Right?

 

But what do these numbers – powers of 10 or orders of magnitude really mean?

Let’s watch the following 1-1/2 minute video to see how these numbers relate to the real world.

Here’s another, more detailed video with explanations. Please watch this video for homework. Answer all the questions following the video.

Questions:1] What is the order of magnitude of the size of;
a] our galaxy
b] the Earth
c] a plant cell
d] the width of a DNA molecule
e] an atom
f] the nucleus of an atom

Part 2: How to calculate using powers of 10

Multiplication is really simple. All you do is add the exponents.
examples;
10^6 x 10^7 = 10^13 (add exponents 6 + 7 = 13)
For negative exponents you still just add but don’t lose the minus signs.
example: 10^4 x 10^-6 = 10^-2 (add exponents 4 + (-6) = 4-6 = -2)

Dividing works the same way except that your bring the power of 10 in the denominator up to the numerator and change its sign. Formally, this is the same as multiplying top and bottom with the power of 10 with its sign changed (reciprocal).

example:  (10^4)/(10^6) =  (10^4)/(10^6) x (10^-6)/(10^-6)

= (10^4 x 10-6)/(10^6 x 10^-6)

numerator: (10^4 x 10-6) =  10^-2

denominator: (10^6 x 10^-6) = 10^(6-6) = 10^0 = 1

So the answer is 10^-2  – in decimal notation this is 0.01

Now do these problems:

a] 10^2 x 10-^2 = ?

b] 10^2 x 10 ^8 =?

c] 10^7 x 10^8 =?

d] 10^-7 x 10^8 =?

e] 10^7 x 10^-8 =?

f] 10^7 / 10^8 =?

g] 10^1 / 10^8 =?

h] 10^0 /10^8 =?

i] 1/ 10^8 =?

Next Lesson: Significant Figures