Types of Infinity

Infinity, cannot be treated as a number, but rather a concept in the sense that it is without any limit. Georg Cantor a famous Mathematician used sets to show how some infinities are larger than others. He did this by showing how a bijection cannot be shown therefore the cardinality (size of the set) must be different between them.

Given two sets A and B, a bijection is said to be true if:

1)There is an Injection between set A and set B (One to one mapping).
2)There is a Surjection between set A and set B (All elements of set A are mapped to an element of set B).

Therefore by proving a bijection between two sets we can conclude that they have the same cardinality.

For example:

Set A (1,2,3,....) and set B (2,4,6,....) can be proved to have the same cardinality. This is because using the function (f(x)=2x), we can create a bijection between set A and set B.

There is a one to one mapping because 2x only has one value.
All elements of set A are mapped to an element of set B because the elements of B are double that of A!

Therefore we have shown that the size of the set 1,2,3,4,5,..... is the same size as the set of 2,4,6,8,10,.....

These are known as countable infinities, because hypothetically speaking, the only thing that stops you from not being able to count all of the numbers 1,2,3,4,... is the time restraint. It is countable because if you counted forever, you could not have missed any numbers between 1 and n, which makes it listable/countable.

However you cannot list the uncountable infinity, because there is no order in which numbers can be arranged, that once you had counted forever, you would have missed no numbers. And for that reason this type of infinity is of different size to the cardinality of the set of natural numbers.

Exams Approaching - Revision Tips

1. Don't overdo it;
The best way is to plan your time efficiently and go for a mix between quality and quantity.

2. Everyone revises different
You need to find the way that revision best suits you; whether it be mind maps, or spider diagrams. Don't worry if you are different to the way that your mates study, different things work for different people.

3. Take breaks
It's proven that studying in short 45 minute periods is much more beneficial than hours of solid revision, so make sure you plan your rest time too!

4.Make a timetable
Doing so will allow you to maximise the time in your day and will actually give you more free time. If you continue to muck about with choosing what to study, you are wasting time that collectively adds up to a lot of unused, wasted time.

5. Food, Water, Exercise
Three components that are absolutely crucial to exam success. Without these the body cannot function as well as it could have done, and puts you to a disadvantage. Keep hydrated, full of energy and ready to take on the world!

6. Sleep
Having enough sleep will help you concentrate better and keep your focus while studying.

7. Balanced Life
The more you study the better you'll be. However remember to go out and treat yourself for the hard work you've been doing.

8. Start studying early
The earlier you start, the more time you have to learn all your content meaning, the earlier you start, the better you'll do!

Good Luck!

Manipulation of Logarithms

With Logarithms, there aren't that many ways that they can manipulated. Here are the 6 different rules that will allow you to manipulate most equations:

Where 'b' is the base
Where 'a' is the part in the brackets

1)  The log can be manipulated  to remove the log if it is given a value: here such as 'c'.

Logb(a)=z       Then b^z=a

2)  The Coefficient of the Log can be moved to the power:

Y Logb(a) = Logb(a)^Y

3)  The Log of (ac) can be written as two different Logs; where they are split by an addition sign.

Logb(as) = Logb(a) + Logb(s)

4) The Division of the Log can be changed to a minus sign, keeping the base of the log the same.

Logb(a/s) = Logb(a) - Logb(s)

5) Any Log with the base the same as the part in the brackets will = 1. This is because anything to the power of 1 will equal itself. 

Logb(b) = 1

6) When changing the base of the Log: Rewrite the same thing but change the base to whatever you want; then divide this by Log with the new base, with in brackets the old base.

Where c is the new base

Logb(a) = Logc(a)/Logc(b)

Square Any 2 Digit Number In Your Head

Maths Trick - Squaring any 2 digit number - In your head!

An easy and fast maths trick to quickly calculate any number squared in your head!!


1. If your number is between 1 and 9 then you can just learn the answer from memory.

2. Work out what multiple of 10 is closest to your number.
For example 20 is the closest multiple of ten to 22.

3. Calculate the difference between the multiple of ten and your number.

4. Add the difference to your original number.

5. Multiply the new number by the multiple of ten.
For example 20 x 24 = 480.

6. Add the difference squared to this number (2 squared = 4).
480 + 4 = 484.


Example 1.

Square 42.

1. 40 is the closest multiple of ten.
2. The difference is 2.
3. 2 add 42 = 44.
4. 44 x 40 = 1760.
5. The difference squared is 4.
6. 4 add 1760 = 1764.


Example 2.

Square 67.

1. 70 is the closest multiple of ten.
2. The difference is 3.
3. 67 minus 3 = 64.
4. 70 x 64 = 4480.
5. The difference squared is 9.
6. 9 add 4480 = 4489.