Math in grade school, hs and college.

Sometimes it's better if you can give an example rather than a dictionary definition. In the case of logs, one application is human hearing. Humans hear pitch change as a ratio, not as an absolute difference. That is, if you're listening to an A-440 Hz, an octave up would be A-880 Hz and an octave up from that would be A-1760 Hz (not 1320 Hz). You would plot frequency on a log axis rather than a linear axis to make sense of this. Further, loudness is correlated to ratios of pressure change. Everyone has heard of "dB" or decibel. This is a log measurement scheme. If someone says that the concert was "100 dB", what they really mean is 100 dB-SPL (sound pressure level), or 100 dB greater than the quietest sound a person can hear. I use logs on an almost daily basis.

JimFiore wrote:Sometimes it's better if you can give an example rather than a dictionary definition. In the case of logs, one application is human hearing.

Perhaps, malmo, perhaps. On the other hand if you said that it is a mapping of an arithmetic sequence onto a geometric sequence I'd think you get even more blank stares. The average person has a very hard time with exponents and doesn't understand the terms arithmetic sequence and geometric sequence, let alone a mapping between the two. I think the example of pitch that I gave (using ratios rather than absolutes) is something non-math people can understand. That is, the idea of the pitch difference between 50 Hz and 100 Hz being the same as that between 5000 Hz and 10000 Hz quickly leads to the conclusion that 10 Hz makes a lot of difference on the bass end but not so much at the treble. Ratios are something people can grasp from everyday experience as in the difference between making $10k/year vs. $20k/year and $200k/year vs. $210k/year.

One of my favorite applications of logs is the use of the log scale for stock price charts of solid public companies. Oh what beautiful diagonal lines some businesses have:

http://finance.yahoo.com/q/bc?t=my&l=on&z=m&q=l&p=&a=&c=&s=expd+adp+payx

malmo wrote:

JimFiore wrote:Sometimes it's better if you can give an example rather than a dictionary definition. In the case of logs, one application is human hearing.

Actually define decibels using logs and pH using logs doesn't do much either. "What do you mean it's a gazillion times louder than quiet? My amp only goes to 11."

How about... it's a method to convert a multivariable exponential function into a linear function. This provides a simplistic view which enables a person to determine incremental changes in their respective problem.

In pH's say a person has a strong acid and a strong base, and this person wants to neutralize the acid. This person starts with the acid and adds specific amounts of the base. By adding a base, the acid will begin to neutralize, but it doesnt neutralize at a rate of one drop of acid to one drop of base, rather over time an incremental addition (one drop) of base will quickly affect the acid's pH toward a more neutral state. Once neutralized, any incremental addition (a drop of base added) to the solution will turn the solution into a base very quickly. If a person where to view this graphically that person could determine the volume of base to neutralize said acid without going over.

I'd like to go back to the birthday problem, because I don't think it was answered well enough yet.

Instead of asking what are the chances of at least two people sharing a birthday, let's consider the oppposite situation, where no two people share the same birthday. Then, we can subtract that probability from one, and the result will be the answer we want (because it covers all the possibilities of people sharing birthdays).

Our first task is to count all the ways that the 23 people could have birthdays, where each person's birthday is different. Well, if we put them on a list, we have 365 possibilities for the first person. For each one of those possibilities, we would have 364 possibilities for the second person. We could make a list of all the possibilities (i.e. 1/1 & 1/2, 1/1 & 1/3, . . . 1/2 & 1/1, 1/2 & 1/3, . . .) but we can see pretty quickly that the total number would be 365*364. For each additional person, we multiply again by the remaining possibilities (365*364*363*362* . . . ), a process called a permutation. (the result is an incredibly huge number). This number we could divide by the total ways that the birthdays could be arranged (365 to the 23rd power) to get the probability that no one has the same birthday, which is about .4927, or 49.27%.

To find the chances that at least two people share a birthday, we subtract that probability from all possible outcomes, i.e. 1-.4927=.5073, or about 50%.

fadfsfdfd wrote:

Here's one of the harder math questions I've had:

Explain what "e" is, and why it is important, to a psychologist (family friend) who has never heard of it.

Well, for one it's the sum of the infinite series 1/0! + 1/1! + 1/2! + 1/3! + ... + 1/n! + ...

Also it's the (i*pi)th root of -1. And the limit as n goes to infinity of (1+1/n)^n.

It's important because it is an exponential function that has many practical applications. Such as: say you wanted to pick the most beautiful woman out of 100, but you only got to see one at a time, and you had to either accept or reject each one when you see her. How many should you "pass" before picking the first one that is more beautiful than all the others to maximize your chances of picking the most beautiful one? The answer turns out to be 100/e.

e is the number which has the unique property that is its own derivative when raised to a power

it is enormously helpful in just about any branch of math including statistics, analysis, and probability to name a few

it is also a cornerstone of one of the most important formulas in math, Euler's formula which states that

e^i*X = cosX + isin X and this in turn leads to the equation that that relates the 5 most important numbers in math(e, i, pi, 1, and 0):

e^i*pi + 1 = 0

OH. Ok.

My favorite application of logs is in the fireplace in winter. Actually, that seems to be the same place lots of folks here want to toss their math books.

So, the first thing i said was, e^2*i*pi=1, but then the question was "what is i..."

Logs were hard to explain. Basically, the way i thought of them when i was little was that the log (10) of a nubmer is the number of zeros it has. So really really big numbers have kind of big logs.

Examples from calculus are kind of meaningless to someone who doesn't know calculus :)

I think the best thing is to define it in terms of limits... like interest compounded continuously. Most people understand money :)

I liked the example of beautiful women, my favorite by far :)

It is a case of people not reading the question correctly and/or applying erroneous methods.

for part i) ie probability of at least two of 23 people in a room having same birthday, that has been answered well enough ie P(A) = 1 - 365*364*...*(365-n+1)/(365^n)

sub in for n=23 gives P(A)=0.507...

Part ii) if A is the event of someone else having the same birthday as you. It is easier to calculate (not A). There are 364 ways that a single person can have a birthday different to your own.

P(A) = 1 - (364/365)^n

Let P(A)=0.5, therefore n=253

Here's a good one:

You have two strings, each one has exactly 60 minutes of "life" before it is burned away when you light it. You're given a book of matches and told to say when 45 minutes is up. How can you do this task?

gausskf wrote:

Here's a good one:

You have two strings, each one has exactly 60 minutes of "life" before it is burned away when you light it. You're given a book of matches and told to say when 45 minutes is up. How can you do this task?

Allow me to clarify. The strings don't necessarily burn at an even rate or at the same rate as each other. And the 60 minutes refers to when you light the string at one of its ends.

good one, the way i think it works is as follows

arrange the 1st string in a circle and the 2nd string in a straight line, touching where the two ends of the first string meet, something like this:

O----

you can now light string 1 in two places and string 2 in one place at the same time

when string 1(the circle) has finished burning string two will have 30 minutes left and will only be burning in one direction, to get it to burn out in 15 minutes light string two from the other end

that will get you darn close to 45 minutes, allowing you a couple seconds to light the other end of string two after one has burnt itself out

Light one at both ends the same time you light one end of the other one. After a half hour the string with two burning ends will extinguish itself, then light the other end of the remaining string and it will burn out in 15 minutes - total time 45 minutes.

That's it. Now for the following:

You and 99 others are captured by a hostile tribe of savages. They then tell you and your fellow prisoners that they will give you a chance to save yourselves. The challenge will involve them sticking either a black or a red spot on each of your foreheads. You will then be lined up single file, and everyone will have the chance to say either "black" or "red," going down the line in order. No other communication is allowed. If you say the color of the spot on your head, you are allowed to go free. You can see what color everyone else's spots are, but not your own. Before the spots are placed on your heads you and your cronies have a strategy meeting. How many people can you save, and how do you do it?

If you can choose which order you can stand in line it is easy. You need three people. We will call one of them the observer, one the asker and the other the passive. The asker should ask (hence the name) the observer if he has the same colour spot as the passive. Then everyone rotates a position ie asker becomes passive et cetera. Then you move on down the line to the next group of three people. Continue until all people know if they have the same colour spot to the person next to them.

You can save all 100.

Surely I've interpretted the vaguely worder question wrongly, a 6 year old could give the same answer that I did.

Notice I wrote "No other communication is allowed." You don't move around in line either. They line you up, you say one word, "red" or "black." You don't get to choose the order. If you say "red" and you have red on, you go free. If you say "black" and you have black, you go free. Otherwise they eat you.