Number
Lesson Objectives
- Understand the difference between natural numbers, integers, common factors, common multiples, rational numbers, irrational numbers and real numbers
- Understand the use of these different numbers, especially for the purpose of the examination.
Whole Numbers
Whole numbers are the positive numbers that do not have a decimal attached with them, and are not in the form of a fraction.
- For example
- 100, 28252 etc.
Whole numbers also include the non-negative integers, which means that Whole numbers include 0.
Thus, whole numbers are 0, 1, 2, 3 and so on to infinity.
Integers
An integer is a whole number. In other words, an integer is a number that has no decimal values attached.
There are three sub-types of integers.
- Positive Integers
- Positive Integers are the whole numbers greater than Zero.
- Thus, the first 10 positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
- Positive Integers are the whole numbers greater than Zero.
- Negative Integers
- Negative Integers are whole numbers that are lesser than Zero.
- Thus, the firs t10 negative integers are -1, -2, -3, -4, -5. -6, -7, -8, -9, -10
- Negative Integers are whole numbers that are lesser than Zero.
- Neutral Integer
- An integer value that is neither positive, nor negative.
- Zero is the only neutral integer.
- An integer value that is neither positive, nor negative.
Natural Numbers
Natural Numbers refer to all the positive integers.
- Thus, the sub-type of integer identified above as positive integers is also known as natural numbers.
- Zero is not a part of natural numbers. When you add zero to natural numbers, it becomes WHOLE NUMBERS.
Prime Numbers
Prime numbers are numbers that have TWO divisors (two numbers on which they divide completely). One of these numbers is always 1, and the other of these numbers is the number itself.
- The smallest prime number is 2, because 2 divides only on 2 and 1.
- Another example of a prime number is 11, because 11 divides only on 1 and 11.
Special Case
1 is NOT a prime number. The reason is that 1 is only divisible by 1. It has ONLY 1 divisor, whereas to be a prime number, it must have TWO divisors. Thus, 1 is NOT a prime number.
Common Factors
Common factors refers to numbers that completely divide two or more numbers. Thus, the common factors of 10 and 20 are:
1, 2, 5, 10
Because both numbers are completely divided by 1, 2, 5 and 1o.
Although 4 and 20 are factors of 20, they do not divide 10 completely, which that they are not common factors of 10 and 20.
To remind here, factor means a number that divides another number completely, without leaving a remainder.
Remember the tricky part, A factor of a number DIVIDES that number.
Common Multiples
A multiple of a number IS DIVIDED by the number completely. For example, 10 is a multiple of 2, because 10 can be completely divided by 2.
Common multiples are numbers that can be completely divided by 2 or more numbers.
For example, a common multiple of 2 and 5, is 20. The reason is that 20 can be completely divided by both 2 and 5. In fact, 20 is a common multiple of both 2, 5 and 10. The reason is that all these numbers divide 20 completely.
Express With Prime Numbers
In order to express a number in its prime factors, we have to divide the number with the smallest possible prime numbers until only 1 is left.
For example, to express 60 in terms of its prime factors, we will divide it with the smallest prime number that can divide it completely. Here, the smallest prime number here is 2
so 60/2= 30
Again, it can be divided by 2
30/2 = 15
Now we cannot divide it completely by 2, so we take 3
15/3 =5
Now, we cannot divide it by 2 or 3, so we take 5
5/5 =1
Thus, we divided the number 2 time with 2, and 1 time with 3 and 5. Thus, our prime factors for 60 are
60 = 2² x 3 x 5
Highest Common Factor
Highest Common Factor (HCF) refers to the largest common number that completely divides two numbers. In order to find the HCF of two or more numbers, we take the least exponent (power) of only the common factors in all the numbers.
Thus, for example, if we need to find the HCF of 10000, 50 and 250
First, we will find the Prime factors of each
For 10000
10000/2 = 5000
5000/2 = 2500
2500/2 = 1250
1250/2 = 625
625/5=125
125/5 = 25
25/5 = 5
5/5=1
For 50
50/2 =25
25/5=5
5/5=1
For 250
250/2 =125
125/5= 25
25/5=5
5/5 =1
Now, as we can see, 2 and 5 are common in all three. However, we need to take the smallest power of the common values. The smallest power of 2 in all three is 1. while the smallest power of 5 in all 3 is 2. So our HCF becomes
HCF = 2 x 5²
Thus, 50 is the highest common factor of the three numbers. In other words, it is the highest possible number the completely divides 50, 250 and 10000.
LCM
The least common multiple is the smallest number that can be divided completely by the given numbers. In order to find the least common multiple, we take the highest power of both the COMMON and UNCOMMON prime factors of each of the numbers provided.
For example, in order to find the LCM of 86, 102, 145
For 86
86/2= 43
43/43 = 1
(43 is a prime number)
86 = 2 x 43
For 102
102/2=51
51/51= 1
102 = 2 x 51
(51 again is a prime number)
For 145
145/5= 29
29/29 = 1
145 = 5 x 29
(29 is also a prime number)
So now we have a no common numbers here. The highest power of 2 here is 1, the highest power of 5 here is 1 again. The highest power of 43, 29 and 51 are 1 as well. So the LCM is
LCM = 2 x 5 x 43 x 29 x 51
LCM =635970
Thus, 635970 is the smallest number that will be completely divided by 145, 102 and 86.
Real Numbers
Any number that can be presented on a number line. It includes all the rational numbers and all irrational numbers.
Rational Numbers
Rational number occurs when two integers divide. Thus, it is one integer numerator being divided by another integer numerator. Thus, any number expressed as p/q, where both p and q are integers, and q is not equal to zero are rational numbers.
- Examples include all decimal numbers that are either recurring or non-recurring.
- All integers, as an integer like 2 or 1000 can be expressed as 2/1 or 1000/1, fulfilling the requirement of p/q
- At O Level, any number that is not a non-recurring, endless decimal, is a rational number.
Irrational Numbers
Any real number that cannot be expressed in fractional form is irrational. At O Levels, there are two main types of irrational numbers.
- Square roots of non-perfect square numbers (covered in next lesson). For example the square root of 2 or the square root of 103 etc.
- A symbolic number, such as pi or π. The reason is that these are endless decimal numbers that are NOT recurring.
- Recurring decimals can be expressed in fractional form, thus, they are rational numbers not irrational numbers.
Lesson Tags
Number | Whole Numbers | Prime Numbers| Detailed Notes For Preparation & Revision | O Level Mathematics D 4024
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