Number - Mathematics D

Number
Cambridge O Level Mathematics D (4024)


This chapter would be quite long and might get boring at times as you might know most of these things. I would rather suggest you to go thoroughly through the number system and give the rest of  the chapter a quick read and update your what you've learned if you find anything unfamiliar.





Number systems:

Definitions of different types of numbers we use:

• Natural numbers: Natural numbers are counting numbers. Natural number include whole numbers only, however, it may or may not include 0 (zero). Natural numbers only include positive numbers.  E.g. {1, 2, 3, 4, 5,...}.
• Whole numbers: All non-fractional numbers are called whole numbers. It includes zero. E.g. { 0, 1, 2, 3, 4,...}. Any number like 0, 7, 212, 1023 are all whole numbers.
• Integers: Integers include all negative and positive non-fractional numbers (or whole numbers) and 0 (zero). E.g. {...-3, -2, -1, 0, 1, 2, 3,...}
• Rational numbers: Rational numbers are all numbers that can be expressed as a ratio or a fraction. This would be possible if both the numerator and denominator are whole numbers or more specifically integers. E.g. { 9.0, 8/1, 2/3, 0.55555} are examples of rational numbers. NOTE: If you put 2/3 in a calculator, you would get the number 0.66666666 and it would go on forever. These are also rational numbers and are expressed using a recurring decimal. A recurring decimal is a decimal number that has digits that repeat forever. 
• Irrational numbers: An irrational number is a number that cannot be expressed as a fraction. This is because in these numbers, there is not a finite number of numbers when written as a decimal. Instead, the numbers go on forever without repeating. Hence, these numbers can't either be expressed using a recurring decimal. E.g. { √2 = (1.41421356…), π = (3.14159265…)}
• Real numbers: A real number is any quantity on the number line. It includes all of the types of numbers mentioned above. E.g. {0, 1, 2, 2.5, 2/3, √2, π}. (√(−6), negative root of something is NOT a real number).
• Imaginary number: An imaginary number is a number which is not a real number. E.g. Negative root of something (√(−6)).

For the symbols of these numbers, refer to your math D syllabus, page#22.





Now take a look at the diagrams to review what you just learned.

I have taken these two pictures have been taken from the internet. They are great for organizing our understanding of number systems




Factors and Multiples:

Factors: The numbers by which another number is divisible are called factors of that number.
E.g.
12 
1 x 12
2 x 6
3 x 4         Therefore, 1, 2, 3, 4, 6 and 12 are factors of the number 12.


Multiples: The multiple of a number is the result of multiplying that number with an integer.
E.g. The first 4 multiples of 6 are: 6, 12, 18 and 24.
6
1 x 6 = 6
2 x 6 = 12
3 x 6 = 18
4 x 6 = 24

Highest Common Factor (HCF):
To find the H.C.F. of two or more numbers, start dividing with the smallest possible number such as 2. If the two  numbers are divisible by two, divide the numbers and write the values in the next lie. Keep on dividing by 2 until either one of the two  numbers are no longer divisible by 2. Then try a larger number 3 if needed proceed in the same way. Stop when you get two numbers that can no longer be divided by a single number. We use the same method for more than two numbers.

Woah! That was quite an elaborated explanation but believe me, this is the best way. Let's go through a few examples to see what actually happens.

HCF of 14 and 32.

14 and 32 are both divisible by 2. Hence we write 2 on the left and 7 and 16 on the next line.
Now, is 7 and 16 divisible by 2? No. By 3? No. By 5? No. If you keep on checking in this way til 7 you would see that there is no other value that can divide both 7 and 16 without a remainder.

Note that I did not check for 4. This is simply because if it is not divisible by 2, how can it be divisible by 4!

Hence, our HCF = 2.



H.C.F. of 64 and 68
We can divide both 64 and 68 with 2 and we get 32 and 34. We can again divide both with 2 and we get 16 and 17. 16 and 17 is not divisible by any single number, hence, we stop.

Our HCF = 2 x 2 = 4

Let's try one with 3 values now.


HCF of 9, 18 and 42. 

We can't divide using 2 as 9 is not divisible by 2 so we go to 3. 3, 6 and 14 are not divisible by any single number so we stop here.

HCF = 3


Lowest Common Multiple (LCM):
In LCM we have to turn whatever number we have into ones. Lets head straight to the examples.

LCM of 14 and 32

Hence, LCM = 2 x 2 x 2 x 2 x 2 x 7 = 224











LCM  of  9, 18 and 42

Hence LCM = 2 x 3 x 3 x 7 = 126












Prime Numbers
A prime number is a number that has only two factors: 1 and itself.
Examples of such numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, etc.

Note that "1" is NOT a prime a number because it has only factor: 1.

Composite number is a number that contains more than 2 factors.
Examples include 4, 10, 15, etc.
The factors of 4 are 1, 2, and 4.

Expressing a number as a product of it's prime factor: 

  Hence 72 = 2^3 x 3^2