CAPE Maths study notes

ok first off here is the syllabus for pure unit 1: http://www.cxc.org/SiteAssets/CAPE_Pure_Math_UNIT1_June_2007_Syllabus.p…
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The Real Number System

Syllabus Requirements:

1.Axioms of the system, i.e. communicative, associative and distributive laws; non-existence of the multiplicative inverse of zero.
2. The order properties
3. Operations involving surds
4. Methods of proof- direct, counter-example
5. Simple applications of mathematical induction

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Revision notes:

The Basics: aka The CSEC stuff you forgot once you left the exam room...

Natural or Positive integers (N/Z+): 1,2,3,4,...
-these numbers are always whole numbers. Notice the lack of a zero.

Integers (Z): ...-4,-3,-2,-1,0,1,2,3,...
-these include both positive and negative numbers

Whole numbers: 0,1,2,3,4,5,...
-this group includes the zero.

Rational Numbers (Q): 2/3, 4/5, 2/1,...
-any number that can be expressed in the form a/b is called a rational number as long as the denominator is not zero. Integers are included because they can be expressed over 1.

Irrational Numbers (Q'):eg.√2
-cannot be expressed in the form a/b.

Real Numbers (R)
-includes rational and irrational numbers and completes the number line.

N.B: a hint that a number is rational is when plugged in on a calculator it will have a finite number of decimal places or will recur. An irrational number will have a infinite amount of places without recurring.

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(1)Axioms of the system

Communicative Laws:
-The communicative law for an arithmetic operation deals with the order in which the operation is performed.
eg. 2+6+9 = 9+6+2 and 2x3x5 = 5x3x2
No matter the order the answer will be the same. Therefore addition and multiplication is communicative.

However, 7-2 is not equal to 2-7 and 8÷2 is not equal to 2÷8. Therefore subtraction and division are not communicative.

Associative Laws:
-This deals with the grouping of arithmetic numbers.
eg. 3+4+7 = (3+4)+7 = 3+(4+7)= 14 and 2x4x5 = (2x4)x5 = 2x(4x5)=40
How the numbers in the operation are grouped does not change the answer. Therefore, addition and multiplication are associative.

However, 9-5-2 = (9-5)-2 but is not equal to 9-(5-2) and 8÷4÷2 = (8÷4)÷2 but is not equal to 8÷(4÷2). Therefore, subtraction and division of numbers are not associative.

Distributive Laws:
-Deals with the multiplication of numbers in brackets.
eg. 3 x(4+7) = 3 x 4 + 3 x 7 = 12 + 21 = 33 and 4 x (8-3) = 4 x 8 + 4 x (-3) = 32 - 12 = 20.
Therefore, multiplication is distributive with respect to addition of numbers and the subtraction of numbers.

Multiplication by Zero:
-any number multiplied by zero is equal to zero.

Inverse of Numbers under addition and multiplication:
-the inverse of a number under a given operation combines with the number under the operation to give the identity.
eg 1. The inverse for 5 under addition is -5:
5+(-5)= 0 (zero is the identity for addition)

eg 2. The inverse for 5 under multiplication is 1/5:
5 x 1/5 = 1 (one is the identity for multiplication)

well thats it for today. The others will be up soon. If any one wants to start the other sections like Algebra and the like go ahead and share. Those who don't have the syllabus go to the link I gave and get it! It's a good way of reminding yourself of what you should know :). Any maths questions you stumbled upon are also welcomed.

Hi queen of the saiyans,

This is really great that you are posting study notes for other exam candidates doing your subject. Bet they will be hooked to your blog now! ;-)

Really, really great! Good luck with your exams! And I know that writing out your notes, helps to commit them to memory so its good for you all 'round.

Holla and let me know if there is anything you need for your blog or discussion forum.

Please no one ask for past papers on this thread that's not what it's for. Just go to my blog and it will give you a link that will take you to a site where you can purchase pastpapers. If you guys really want to pass you'll put your hand in your pocket and bloody well buy the books. [I mean that in the nicest way possible :)]. I'll put up some questions that can be used in revision also.

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Ordered Properties

Real numbers have the property that they are ordered, which means that given any two different numbers we can always say that one is greater or less than the other. A more formal way of saying this is:

For any two real numbers a and b, one and only one of the following three statements is true:

1. a is less than b, (expressed as a < b)

2. a is equal to b, (expressed as a = b)

3. a is greater than b, (expressed as a > b)

The ordered nature of real numbers lets us arrange them along a line (imagine that the line is made up of an infinite number of points all packed so closely together that they form a solid line). Each Real number corresponds to exactly one point on the number line. The points are ordered so that points to the right are greater than points to the left. This explains Negative numbers which are to the left and decrease in value the further they are away from zero and positive numbers which are always greater than negative numbers. The arrow at the end of the number line shows that the line leads to infinity on both sides.

Here's a slide show: http://www.slideshare.net/rfant/properties-of-real-numbers

Oh hey thanks a lot admin. You're right it does help me revise. I really wanted to do it because I was tired of going to a thousand sites just to get information for pure maths. It gets very frustrating after a time :)

http://teachmitv.wordpress.com/category/resources-tools/cape-tutorials/

This link will take you to a site that teaches both csec and cape maths through videos. They go through almost every topic.

Another site is examsolutions: http://www.examsolutions.co.uk/A-Level-maths-tutorials/index.php
The teaching is clear, concise and you'll cover a huge amount in very little time. It's a really great site for revision. It covers gcse (CSec students take note), the core of A-level maths (C1-C4), some M1 for mechanics and some S1 and S2 topics for statistics.

Also take a look at this guy's videos for revision on Calculus and some math topics: http://www.youtube.com/watch?v=u06Yrvt2XLc&feature=related
I find his videos very easy to understand.

However, if you want to start from the very beginning of Calculus then this is the site for you: http://www.rootmath.org/calculus1/welcome-to-calculus1.php

I'll put more as I find them. Happy revising! :)

I really must comment on this site again I find it extremely good. I will admit to having learnt most of my math here. The videos are clear, concise and more importantly: he also covers Applied maths. The videos are mainly focused on answering Edexcel pastpapers and the key points the student must remember in order to get full marks. I don't really care about the board its for only the fact that the content is exactly what I needed to know. I recommend this site for anyone serious about passing Pure or Applied Math. Good luck everyone!

http://www.examsolutions.co.uk/A-Level-maths-tutorials/maths_tutorials_…

Will be putting other exam sites and notes fairly soon so look out for it!

Where can I get videos that teaches Cape Integrated Mathematics