Measuring angles - why 360 degrees?

Most of the time at school we use 'degrees' to measure an angle... first of all what is an angle???
I think of an angle as the amount you have to turn one line so it meets the other.

There are lots of ways of measuring this but a very popular one is using degrees, where we say that there are 360 degrees in a full turn. But why do we use 360? Why not 100? or 400?

It seems to go back to the ancient Babylonians (in an area which is now within Iraq), who were extremely influential on a lot of maths we still use today, as have been many other cultures such as the Greeks, Chinese and Indians.
The Babylonians counted using 60s rather than tens and considered the number 6 to be of strong importance and therefore 6 x 6 = 36 to be particularly significant. This was supposedly backed up by observing there are approximately 360 days in a year, so a full circle being made up of 360 parts or degrees seems to make sense.
360, by a bit of luck(?) turns out to be fairly useful because it has a lot of factors 

However there is another useful way of measuring angles using 'radians' instead of degrees.
Radians are particularly useful for higher level maths and are based on the idea of how many lots of the diameter of a circle, with radius of 1, will wrap around the circumference.
1 radian is equal to about 57.2958 degrees. Radians use the value π 

x.... why do we use x in Maths?

In Maths when we want to use a letter to stand for a number that we don't necessarily know yet or that we want to use different values for... we often use the letter x ... for example... the cooking time for a chicken is 40 minutes for every kilo, so the time it takes depends on how heavy the chicken is... 
We could turn this into a formula where  x  is the weight (in kg) of the chicken  Time = 40x
This is what we often mean by algebra

So...
when x = 1 the cooking time is 40 x 1 = 40 minutes
when x = 2 the cooking time is 40 x 2 = 80 minutes
when x = 3 the cooking time is 40 x 3 = 120 minutes
when x = 4 the cooking time is 40 x 4 = 160 minutes
when x = 5 the cooking time is 40 x 5 = 200 minutes
....

The formula Time = 40x   saves us from having to keep writing this out lots of times!

But why x?

There are a few theories however a popular one is that back in the 17th Century in France a famous philosopher and mathematician René Descartes was sending a lot of maths to get get printed. In order to print a book the printers had to use individual blocks of wood with each letter. In René's maths he used lots of different letters to stand for the 'missing numbers' and the printers suggested that in the French language they didn't use many x's so they had plenty of them they could use.... so as a result, all of René's 'missing numbers' are shown using x.

René Descartes also introduced the idea of using (x,y) coordinates and graphs

Click here to read more about the history of algebra and here

Number patterns and the Fibonacci sequence

Maths is all about discovering patterns, trying to find out how they work and using them to help us.

Sequences are examples of number patterns and they can be pretty straight forward such as the 2 x table:

0, 2, 4, 6, 8, 10, ...

or they can be more complex such as:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

This sequence is really interesting and it turns up all over the place, especially  in nature. It was first explored by Leonardo of Pisa, nicknamed Fibonacci.

Find out what it has to do with...

Rabbits         Petals and Sunflowers        Sea Shells    and more... even more...

Year 12 further maths - Complex Numbers

The 'invention' of complex numbers isn't that old and people still find it hard to believe that they aren't just made up nonsense!

"A complex number is a number that can be put in the form a + bi, where a and b are real numbers and i is called the imaginary unit, where i² = −1.^[1] In this expression, a is called the real part and b the imaginary part of the complex number."

Have a look at the history of them ...http://rossroessler.tripod.com/
Especially the role of some mathematicians in 16th century Italy... http://env1.kangwon.ac.kr/cyberclass/undergrad/na/2008/website/numerical/ComplexNumberOrigin.html

...and have a look at some of the uses of them in our very real world http://en.wikipedia.org/wiki/Complex_number#Applications

exponentials and logarithms

In Year 13 we're looking at e and ln

e and ln are useful for describing many processes in the natural world and being able to integrate and differentiate functions involving them enables us to understand how things in the natural world behave.

http://betterexplained.com/articles/using-logs-in-the-real-world/

http://algebralab.org/lessons/lesson.aspx?file=Algebra_Exponents_e.xml

http://en.wikipedia.org/wiki/Natural_logarithm

Check the MyMaths lesson http://www.mymaths.co.uk/tasks/library/alevel/lib/loadLesson.asp?title=alevel/core3/IntegrationC3/integration9exlnx&taskID=2168

and check your understanding http://www.mymaths.co.uk/tasks/library/alevel/lib/loadTask.asp?title=alevel/core3/IntegrationC3/integration9exlnxOH&taskID=2168

Have a read around and find out where else lograthims and exponentials crop up... and post them here :)

Twitter...

If you're on twitter then follow me @MrJamesKCC

New beginnings 2, 3, 5, ...

The new school year has started and this year I'm hoping to help you to find out how amazing maths is and how the maths you learn in school is used in the world around us. It's not always easy in school to show or look at all the interesting maths in the world and hopefully this blog will try to show you.

I'm hoping everyone from Year 7 to Year 13 will be able to use the blog but don't be put off if the things you look at seem too complicated, sometimes things aren't as complicated as they seem. All mathematicians start off not really understanding what's going on but with a bit of effort, determination and interest they begin to understand and appreciate how amazing our world is... All by using maths!

I hope you enjoy it and if you find anything you think other people might find interesting let me know!

To get started have a look at this about some of the most useful numbers... Primes! http://www.murderousmaths.co.uk/games/primcal.htm