Numbers, Squares, Square Roots, Cube, Cube Roots. | O Level Mathematics 4024 & IGCSE Mathematics 0580 | Detailed Free Notes To Score An A Star (A*)

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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.

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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.
• 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
• Neutral Integer
  • An integer value that is neither positive, nor negative.
    • Zero is the only neutral integer.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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Real Numbers

Any number that can be presented on a number line. It includes all the rational numbers and all irrational numbers.

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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.

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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.

Directed Numbers

Directed numbers are those numbers that have a size and a direction. The numerical part of the number shows the size, while the sign with the number sows the direction.

• An example of Directed Numbers is temperature, where a negative temperature means temperature below zero, while a positive temperature means a temperature above zero. Thus, + 25°C means the size of the temperature quantity here is 25 and the direction here is positive or greater than zero.
• Always remember that a number that has no sign attached with it is a positive number.

How To Count Directed Numbers?

A major mistake that students incur when counting directed numbers is to miss out Zero.

• For example, if you are asked that the temperature today is 10°C in the morning, and the temperature fall by 12 degrees, what was the temperature at night time after the fall?
  • When we count back, we realize that
  • If we do not count zero in our calculation, the final temperature becomes -3°C.
  • However, this temperature calculation is wrong, as temperature can be 0°C, so we have to count zero.
  • Thus, counting back from 10 with 12 degrees, gives us -2°C temperature after the change has occurred.

1.1 Square Numbers

The product of a number with itself is called the square of the number. In other words, square is the number that is formed when a number is MULTIPLIED with itself. There are a few things you must remember about square numbers:

• A square of a number can never be negative. It is always positive
  • For example the square of 2 is (2 x 2) = 4
  • The square of negative 2 or -2 is (-2 x -2) = 4
    • The two minus multiplied with each other form a positive.
    • You may recall that
      • Negative multiply with negative = positive
      • Positive multiply with positive = positive
      • Negative multiply with positive = negative
    • In other words, if two opposing signs are multiplied together, they end up in a negative result
    • A negative result is one with a minus sign, while a positive result is one with no sign or a plus sign.
• The question asked in the exam may ask you to take the square of a number by inserting a superscript.
  • Always remember, square is denoted by super script 2
    • For example if the examiner wants to ask you to take the square of 2, he may ask
      
    • You must remember that if there is ONLY a number, then the square involves its sign
      • For example
        is the same as (-2 x -2)
• A perfect square is one where the square is formed by multiplying the same integer with itself.  An integer is a number without decimals.
• 0 is not a perfect square. When a question asks you to add the first five perfect squares, you start from 1.

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1.2 Square Roots

Square root is the opposite of square. It is used to find out which number was multiplied with itself to achieve the square that we have. A few crucial aspects to keep in mind are

• A square root is denoted by √
  sign.  Anything inside it is what you need to take the square root of. For example [latex] sqrt 25 [/latex] asks you to find the square root of 5.
• A square root ALWAYS has two answers, one positive and one negative. These are the positives and negatives of the same number.
  • For example, the square root of 25 is 5, however, it is not just positive 5. Instead, negative 5 or -5 is also a square root because -5 x -5 = 25 as well. Thus, our answer for square root of 25 is
    ±5 which represents positive and negative 5.
• At the O level stage, you must remember that the answer for square root of ANY NEGATIVE NUMBER is that there is no square root. Thus square root of -1, -100, -1000 etc. does not exist at this level of student.
• You must remember that square root is calculated using prime factors:
  • What is the square root of 10000
  • First try to find all the prime factors of the number. These are the prime numbers that multiply together to firm the number. Start from the smallest
    • Special aspect to remember, prime factors are numbers that are only divisible by themselves and 1. Also, these are always positive at this level. So the first prime number is 2, as 2 can only be divided completely by 1 and 2.
      • 1 is not a prime number because it does not have 2 divisors. It only has one divisor, ITSELF.
    • If we divide 10000 with 2 we get 5000
    • Again 5000/2 = 2500
    • 2500/2 = 1250
    • 1250/2 = 625
    • 625 is not divisible by 3 or 2, both of which are prime numbers. So we take the next prime number that is 5.
      • Special Tip – a number is only divisible by 3 is all the numbers in it add up to a number divisible by 3
        • So  1082582 = 1+0+8+2+5+8+2 = 26 which is not divisible by 3, so 1082582 is not divisible by 3
        • 129225 = 1+2+9+2+2+5 = 21 which is divisible by 3 ( 3 x 7 =21). So, 129225 is divisible by 3.
    • 625/5 = 125
    • 125/5 = 25
    • 25/5 = 5
    • 5/5 = 1
  • So we have 2, 2, 2, 2, 5, 5, 5, 5 as the prime factors. We can make two equal sets from it
    • 2 x 2 x 5 x 5 and 2 x 2 x 5 x5 are equal sets
    • Thus, our square root for 10000 is
      ±100
• At this stage, in O levels, you are sometimes asked to identify irrational numbers. Pro tip – square root of negative numbers are irrational.
• The first perfect square is 1.

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1.3 Cube Numbers

Cube numbers are similar to square numbers. The main difference is that we multiply the same number with itself 3 times. It is denoted by a 3 in the super script

• Cube numbers can both be positive or negative. The cube of the same number in its positive and negative forms are different. For example the cube of 2 and -2 are NOT the same.
  • The cube of 2 or
    • 2 x 2 x 2
    • 8
  • The cube of -2
    • -2 x -2 x -2
    • Lets do it step wise
      • -2 x -2 = 4
      • 4 x -2 = -8
• The examiner may ask questions like evaluate
  
  • 10 x 10 x 10 = 1000

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1.4 Cube Roots

Cube root is the same as square root, instead that we have to find the number that multiplies 3 times with itself to form the cube number. Thus, the cube root of 1000 is 10. However, it must be noted that there is only one sign, because cubes of same number with different signs have different signs as well.

• Cube root is denoted by
  
• The sign of the answer is extremely important in cube roots.
• The same prime factor method is used to determine cube roots as well.
• Find the cube root of 64
  • 64/2 = 32
  • 32/2 = 16
  • 16/2 =8
  • 8/2 = 4
  • 4/2 =2
  • 2/2 = 1
• We divided 6 times with 2, so we have 2, 2, 2, 2, 2, 2
• We need to to make 3 equal sets: 3 sets of 2 x 2 each.
• So the cube root is 2 x 2 = 4
• IF it was -64, the final answer would have a – as well. So it would have been -4.