Benford's Law

Benford's Law states that if d is the first digit of a number occuring in real life, that digit should occur with probability P(d) = lg (d+1) - lg d. The space between log scales.

P(1) = lg 2 - lg 1 = 0.301
P(2) = lg 3 - lg 2 = 0.176
P(3) = lg 4 - lg 3 = 0.125
P(4) = lg 5 - lg 4 = 0.097
P(5) = lg 6 - lg 5 = 0.079
P(6) = lg 7 - lg 6 = 0.067
P(7) = lg 8 - lg 7 = 0.058
P(8) = lg 9 - lg 8 = 0.051
P(9) = lg 10 - lg 9 = 0.046

What does this imply??? If you were to compile all the numbers in your bank books and if were to take a look at the first digit, they should follow this distribution in the long run. This is a way that IRS have learnt to spot people who fudge numbers in their books. People who fake their numbers tend to distribute their numbers more "randomly" but thats not true in real life.

Its actually quite counter-intuitive as people assume that if numbers are random, they should be equally distributed. However, numbers in real life are not completely random. Numbers starting with '1' occurs more than 6 times more than '9'.

Powers of 10

Its always amusing to see answers (length, speeds, masses) that are rediculously out of proportion. It can be faster than light, denser than neutron stars or heavier than the earth. Thats why its always good to get them to appreciate what difference a power of 10 makes.

There are some good websites and tools that help put large powers of 10 in physical perspective. This can be quite useful in teaching physical units, scientific form and plain old estimation.

Powers of 10 website with the original video

Interactive Java version (Much Smoother)

Say "Principia"

Principia is a word that is usually mis-pronounced by many. This post is to vindicate myself when others have tried to correct me.

Now, we come across this word in Math because mathematicians sometimes refer to either the 3 volume text "Principia Mathematica" that chronicles the foundations of mathematics. (Famous for Whitehead/Afred/Bertrand's proof of 1 + 1 = 2) or Newton's work "Philosophiæ Naturalis Principia Mathematica".

Principia should be pronouced with a strong "c". In latin, the "c" is pronounced more like a hard "K", hence the correct way to say principia in english is prin-KEE-pia.

A Prime Problem

Given that a and b are positive integers and that:Find the possible value(s) of a and b.

Ans

This problem can be given after students have learnt to factorize to challenge them. By now, they will be thinking... Huh? Only one equation, how to solve for 2 unknowns?

But lets try to factorize first. We will get:

(a + b) (a - b) = 97

What students have to realize here to proceed is to recognize that 97 is a prime number.

Hence, a + b = 97 and a - b = 1

Solving those 2 equations give us a = 49 and b = 48

This question can of course be reduced to smaller primes to simplify this problem or even non-primes but that will give multiple pairs of solutions. If I had not stated that a and b must be positive integers, students must also consider for the possibility that a + b = 1 and that a - b = 97 for the negative solutions.

Educational Stories 01 - The Barometer Problem

I'm going to use this blog to also collect some really educational stories which can be used in class to inspire students. Let me begin this series with the famous Barometer Problem which I used in all my physics classes when I lecture on pressure.

During an exam, students were asked to explain how to measure the height of a building using a barometer.

*Now if you are a student of mine, you should be able to answer this after the lesson on pressure but let me just run through the proper "textbook" answer.

Pressure = density x gravitational acceleration x height

So if you measure the pressure difference from the top of the building and below, you can divide it by the density of air and the gravitational acceleration to get you an approximate height difference. This is assuming that density of air and gravitational acceleration is constant over the height of this building.*

Now in this exam, there was a student who looked extremely puzzled/constipated when he was tackling the problem. When the teacher started marking the papers he noticed that boy's answer.

Ans: I would take a string and tie it to the barometer. I will lower the barometer to the ground and measure the length of the string.

DOH! The boy wasn't wrong. BUT he did not demonstrate any knowledge of pressure which was the aim of the assessment. Furious yet tickled at the same time, the teacher summoned the boy to his office.

He asked the boy to re-do the problem but this time to use concepts he thought in class. The boy became more puzzled. He replied: "Which concept? I have many answers!"

The teacher was astonished. He asked the boy "Many answers? Tell them to me."

The boy answered:

"Method 2: I could use units. I walk up the building in the stair case and mark on the walls each time i pass the height of the barometer and count how high the building is in terms of barometers. After measuring the height of 1 barometer with a ruler, I can convert that height form barometer lengths into standard units.

Method 3: I could use Kinematics. I release the barometer from the top of the building from rest and measure how long it takes to reach the ground with a stopwatch. Using the formula taught:

x = ut + 1/2 at^2

I can deduce the height of the building x by substituting u = 0, a = 10m/s^2 and t the time measured on the stopwatch.

Method 4: I could use Period of pendulum by constructing a huge pendulum of length equal to the building and measure the period of the pendulum when swinging it from the top of the building.

T = 2pi root(length of pendulum/g)

Measuring the period, the height of the building could be calculated for the equation above.

Method 5: Similar triangles. I place the baromter standing on the floor and measure the ratio of the height of the baromoter to its shadow. I can calculate the height of the building with respect to its shadow.

Of course, the easiest answer would be to just give the building manager the barometer in exchange of information about the building."

The teacher was dumbstruck and gave the boy an A.

You can find this story in many websites attributing the boy to Niels Bohr but that isn't true. Nevertheless, this makes a great story to inspire students and to round up basic mechanics in Physics or Similar triangles in mathematics.

Mathematical Pi

I thought I would start off my math blog with a Mathematics Song I've always enjoyed entitled "Mathematical Pi". This is a video version from youtube:



The song was initially written Ken Ferrier & Antoni Chan. Here is another version on a website: http://www.vvc.edu/ph/TonerS/mathpi.html

I've used this in class to remind students a few facts about pi as well as take them through a short history of the number Pi.

1. Pi is defined as the ratio of the circumference of a circle to its diameter.
2. Pi is a constant.
3. Pi is NOT 22/7 nor is it just 3.14 or 3.141. These are approximations we use for the sake of easy calculations as they won't affect the accuracy of our answers much.
4. Pi is irrational (i.e. you cannot represent it as a fraction nor can you express it as a recurring decimal.) Well... you can represent all recurring decimals as a fraction.

I recommend using this song as a memory aid if you really want to impress people with how many sig figs you know pi to. (Just don't go challenging Daniel Tammet... his record is 22,514 digits)