Mathematicians - a little crazy?

Sometimes math problems drive mathematicians a little crazy...

Drawing a line (basics)

There are three basic methods that one can use to draw a line in the xy-plane..

1. The table method:
This involves drawing up a table with x and y values.
2. The intercept method:
We find that x- and y-intercepts with this method.
3. The gradient and y-intercept method:
We use the y-intercept and the gradient.

Let's consider an example: y = 2x - 4
Remember the general formula for a line which is y = mx + c. In this case m represents the gradient and c represents the y-intercept.
The y-intercept is where the line intersects the y-axis on the xy-plane.

1. We choose some x values, say -2, -1, 0, 1, 2. We then substitute these values into the equation to obtain the corresponding values of y. We usually do this by drawing up a table to make it easier to see which x and y values correspond.
In our example, if x = 1, then y = -2, so we plot the point (1, -2) on the xy-plane. After plotting a few point you should be able to draw a straight line which goes through all the points.

2. We already know that c represents the y-intercept, which in our case is -4. So we know that (0, -4) is a point on the line. To find the x-intercept we let y = 0, then we find that x = 2. So (2, 0) is also a point on our line. Now we can draw a straight line through these two points and we should obtain the same line as with the first method.

3. We know that the y-intercept is -4. Also, m represents the gradient which is 2. The gradient is also known as the slope and can be found be considering the fraction y/x. So, if y/x = 2, this is the same as saying y/x = 2/1. Hence, we have the point (1,2) on our line, as well as (0,-4). Now we draw the straight line through these points.

The Batman equation

Even Batman can be recreated through math...

Math dance class

(From: http://www.caption-of-the-day.com)

The number 2

Even though 2 is such a small number, there are various properties illustrating the usefulness of this number.

First of all it is the second number in the natural number system.
It is an even number and also the only even number which is a prime number.
It helps us to determine whether an integer number is even, i.e. a number is even when it can be divided by 2.
The Roman numeral for 2 is II.
In some situations 2 can be called a couple or a pair, for instance two socks is called a pair of socks.
If we divide 100 by 2² we arrive at 25, the previous two number which I have discussed (see: The number 25, The number 100).
Being number 2, you can also say being second. In a medal race this would usually mean that you will receive the silver medal.
Number 2 is a fictional character in the Austin Power movies.
In terms of imaginary number we can write 2 as the product: 2 = (1+i)(1-i).
When we draw graphs we usually draw it in a 2-dimensional space, the xy-plane.

If you know of any other interesting facts, which I'm sure there are many of, please leave a comment.

The number 100

A lot can be said about this number. Let's take a look at some of its properties:

- 100 is the first 3 digit number when we count from 0 onwards.
- It consists of only 1's and 0's hence we can also find this presentation in binary numbers where it would represent the number 4.
- It is an even number.
- It's divisible by 1, 2, 4, 5, 10, 20, 25, 50, 100.
- It is equal to 10 squared.
- The square root of 100 is 10.
- It represents a century (in terms of years or cricket).
- There were 100 dalmatians, in the movie 101 Dalmatians, with spots, except for Spotty who only developed his spots at the end.
- It can be represented by the Roman numeral C.
- The year 100 was a leap year!
- If we factorize 100 we have: 100 = 2.2.5.5.
- It is a natural number (hence also rational and real).
- In terms of percentages it represents the full 'amount', i.e. 100% in a test would mean that you answered everything correctly. 100% profit would mean your profit will be equal to the cost price.

For some more interesting facts see: Facts about 100.

Pizza math

Not mathematically correct... But who cares? It's pizza!

Prime numbers

A prime number is a number which can only be divided by 1 and itself. This means that there is no other number which can divide into it such that the result is an integer.

Here are a few examples of prime numbers:
1, 2, 3, 5, 7, 11, 13, 17, 29, 31

The number 2 is the only even prime number, since all other even numbers are divisible by 2.

Very large prime numbers are used in encryption as 'keys' to secure data over networks.

MSc degree in Applied Mathematics

I am finally done with my MSc in Applied Mathematics! It's been almost two years of hard work and dedication and I am relieved that it is done.

Next year: PhD!

Blogging x 0 =Exam

With my final MSc exam a mere three days away, this blog would have to wait until ithe storm has passed before it will once again carry meaningful posts. Here follows the abstract of my presentation:

How fast will my coffee get cold?

With a not so difficult mathematical formula you can calculate exactly at what time will your cup of coffee (or tee) get cold (or to a certain temperature). It all depends on the room temperature, if it's colder the coffee will get colder faster and if it's a hot summer's day you coffee will stay warm for longer.

Let's take a look at the formula:







This is called a first-order differential equation. The T refers to the temperature at any given time t and A is the room temperature. The constant k indicates the rate at which the temperature will change.

Let's solve this equation:


Now we need to A (room temperature), B (which we can find by knowing the initial temperature of the coffee) and k (which is a bit more difficult to find).

Let's say A = 22˚C and that since water boils at 100˚C and we immediately make our coffee, let's assume that T(0) = 90˚C (Taking into account that it will cool off a bit when pouring and stirring).

Now we have the following:

To find the value of k we need to do some experimentation, where we determine the temperature of the coffee after some specific time. So let's say that after t=2 minutes we find that the temperature of the coffee is now 85˚C (I'm just guessing a value, for exact results one would have to take the actual temperature of the coffee after two minutes).

So now we find k:

Hence the equation becomes:

If I now want to know the temperature of my coffee after 20 minutes, let t = 20, then

and you will see that as time goes by the temperature will drop at a slower rate until it reaches the room temperature.

Happy coffee-drinking!

Chocolate math

Love math - Put on your positive thinking hat

I love math. Who doesn't?
If you don't it must be because you haven't met with its amazing properties and applications.

For many people it's understanding math that is the problem and from that the negativity spirals out onto those around them and hence the general dislike of mathematics.

I believe that everyone are able to do math and understand the basic ideas, excluding perhaps all the complicated concepts one might encounter past a first year university level. The problem is the attitude going in, the believe that you cannot do something before you even tried. With that kind of thinking no one will ever get anything done.

Let's all put on own positive thinking hats wherever we go in this world and not only embrace math, but all seemingly difficult tasks that are throw unto our paths everyday. The next time someone wants to teach you something about math, say yes and you might just be surprised.

Our minds can store an endless amount of information so why not use some of that space to learn something new and interesting every day?

Happy positive thinking!

How not to expand an equation

The correct answer would be:


and we can make use of the Binomial theorem to find all the other terms.

Trigonometry Series (Part 4)

Today we look at several formulas which tend to be helpful when we are solving trigonometric equations or working with specific angles. They are also useful for simplifying expressions.

Addition formulas:


The formula for tan can be derived by making use of the formulas for sin and cos.


Subtraction formulas:

 
The subtraction formulas can be easily derived from the addition formulas by making use of the angle identities in Part 3.

Double-angle formulas:


These formulas can be derived from the addition formulas by letting y=x and using some identities.

Half-angle formulas:



These formulas can be obtained by rewriting the double-angle formulas.

Next time we will consider a few problems and examples, showing how we can apply and use all the information that we have from the previous parts in the series.