- Numbers can be classified into Real numbers and Complex numbers
- Complex numbers are in the form of a+ib, where ‘a’ is real part and ‘ib’ is imaginary part. ‘i’ is equal to square root of negative 1. Therefore, ‘i-squared’ is equal to -1.
- Real numbers can be classified into Rational numbers and Irrational numbers.
- Rational numbers are numbers which can be written in the form of p/q, where p and q are integers and q is not equal to zero. Examples: 1/3, 3/1, 3/2, 3/3, 0/1, 2/3.
- Irrational numbers are numbers which cannot be written in p/q form. They are non-recurring and non-terminating. Examples: √2, √3, π
- Integers are sub set of rational numbers. They can be written in the p/q form. However, q is always equal to 1. Thus 0/1, 1/1, 2/1, 3/1 are all integers. They are equal to 0, 1, 2, 3 respectively. In addition, it includes integers in negative form. -1, -2, -3 are also integers. They are called negative integers.
- Numbers of the set 0, 1, 2, 3, and so on are called whole numbers.
- Numbers of the set 1, 2, 3, 4, and so on are called natural numbers. These are the numbers we use to count physical objects.
- Numbers can also be classified as ‘odd’ and ‘even’. 1, 3, 5, 7, and so on are called odd numbers. 2, 4, 6, 8, and so on are called even numbers. Zero, by definition, is also an even number. An even number is a number, which can evenly be divided by 2.
- Numbers can also be classified as prime and composite numbers
- Prime number is a number, which can be evenly divided only by the number 1 and the number itself. Examples: 2, 3, 5, 7, 11, 13 and so on. 1 is not defined as a prime number.
- Numbers which are not prime are called composite numbers. They have more factors other than number 1 and the number itself. Examples: 4, 6, 8, 9, 10, 12 and so on.
- If two natural numbers have no common factors between them except the common factor ‘1’, then they are called co-primes. For example, 3 and 4 or 5 and 6 or 5 and 7.
- Suppose ‘f’ is a number and ‘m’ is another number. If ‘f’ divides ‘m’ evenly, then ‘f’ is called a factor of ‘m’ and ‘m’ is called multiple. Factors are also known as divisors. Example: 2 is a factor of 4; 2 and 3 are factors of 6. Also note that 4 is a factor of 4 and 6 is a factor of 6.
- All the even numbers are divisible by 2.
- If sum of the digits is divisible by 3, then the number is divisible by 3. For example, the number 123 is divisible by 3, because 1+2+3 is divisible by 3.
- If the last two digits is divisible by 4, then the number is divisible by 4. For example, 2916 is divisible by 4, because 16 is divisible by 4.
- If the unit digit is either ‘0’ or ‘5’, then the number is divisible by 5.
- If the number is divisible by 2 as well as by 3, then the number is divisible by 6. For example, 126 is divisible by 6, because it is an even number and also divisible by 3.
- If the last three digits are divisible by 8, then the number is divisible by 8 because 1000 is always divisible by 8. For example, 1816 is divisible by 8 (as 816 is divisible by 8).
- If sum of the digits is divisible 9, then the number is divisible by 9. For example, 126 is divisible by 9, because 1+2+6 is divisible by 9.
- If a number is divisible by 3 as well as by 4, then the number is also divisible by 12. Note that 3 and 4 are co prime numbers. This is an essential condition for this divisibility. Therefore, this can be generalized to other composite numbers. For example, 18=2*9, where 2 and 9 are co prime numbers. Therefore, if the number is divisible by 2 as well as by 9, then the number is also divisible by 18.
Divisibility condition for certain prime numbers such as 7, 11, 13, 17, 19, 23, 29, 31, 37, and 39.
- When 1000 is divided by 7, the remainder is 6 or -1. When 100 is divided by 7, the remainder is 2. These information can be applied to find the remainder when a number is divided by 7. For example, the number 86,415 is divisible by 7. The number contains 86 thousands. Therefore, the remainder of 86,000 is -1*86=-86. Now, add this number to 415; you get, 415-86=329. Since, 329 is divisible by 7, the number is divisible by 7. Or, -1*86= -1*2=-2. -2+415=413. 4*2+13 = 21. As 21 is divisible by 7, the number is also divisible by 7.
- Split the given number into two sequences: Odd placed and Even placed. Find the sum of each sequence. Then, find the difference of these two sums. If it is either zero or multiple of 11, then the given number is also divisible by 11. For example, 2431 is divisible by 11, because (2+3)-(4+1)=0; 9020 is divisible by 11, the difference is 11.
- When 1000 is divided by 13, the remainder is 12 or -1. This is similar to the rule of 7. Note that you can use the remainder ‘-1’ for both 7 as well as 13.
- When 1000 is divided by 17, the remainder is -3. 2006 is divisible by 17, because 2*-3 + 6 = 0. 4012 is divisible by 17, because 4*-3 +12 = 0.
- When 100 is divided by 19, the remainder is 5 and when 1000 is divided by 19, the remainder is -7. These two information can be used for divisibility of 19. For example, 86108 is divisible by 19. 86108=86000+100+8. (86*-7)+5+8=-589=600-11. (6*5) – 11 = 19. Therefore, the number is divisible by 19. Alternatively, when 400 is divided by 19 the remainder is 1. There are 86 thousands or 215 four hundreds. Therefore, 215+108=323=3*5+23=38. 38 is divisible by 19.
- When 300 is divided by 23 the remainder is 1. For example, when 86108 is divided by 23 the remainder is 19. There are 861 hundreds or 287 three hundreds. Therefore, 287+8=295. When 295 is divided by 23 the remainder is 19.
- When 2000 is divided by 29, the remainder is -1. Therefore, 2001 is exactly divisible by 29 (-1+1=0). Let us see what’s the remainder when 86108 is divided by 29. It has 86 thousands or 43 two thousands. Therefore, -43+108=65. 65-58=7. Hence, 7 is the remainder.
- When 4000 is divided by 31 the remainder is 1 and when 400 is divided by 31 the remainder is -3. For example, when 86108 is divided by 31 the remainder is 21. 86108 = 84000+2000+108. 21+5*-3+15 = 21.
- When 1000 is divided by 37 the remainder is 1. For example, when 86108 is divided by 37 the remainder is 9. 86+108=12-3=9.
- When 40 is divided by 39 the remainder is 1. When 1000 is divided by 39 the remainder is 25. For example, when 86108 is divided by 39 the remainder is 35. 86*25+108= 8*25+30=200+30=5+30=35.
Number of factors of a composite number, N
If N is a composite number and N=a^p*b^q*c^r……, where a,b,c are distinct prime factors of N and p,q,r are positive integers, then the number of factors of N is (p+1)(q+1)(r+1)….. For example 180=4*9*5=2^2*3^2*5^1. The number of factors of 180 is (2+1)(2+1)(1+1)=18
Using this formula, we can also get the number of ways of expressing a composite number as a product of two factors. For example, 40=5*8=5^1*2^3. Therefore, the number of factors is 8. Hence, we get 4 pairs of factors that can be expressed as product of 40:1*40, 2*20, 4*10, and 5*8.
However, if the number is a perfect square, then we have to adjust our result for the pair, which contains the same factor. For example, 36 is a perfect square. The number of factors is 9. This is not evenly divisible by 2. So, we have to add 1 to 9 and divide by 2, which gives us the answer 5. In fact, they are right:1*36, 2*18, 3*12, 4*9. 6*6.
Sum of the factors of a composite number, N
We can also find the sum of all factors of a composite number, N. The formula is:
(a^(p+1)-1)/(a-1) * (b^(q+1)-1)/(b-1) * (c^(r+1)-1)/(c-1)
For example, 36=4*9=2^2 *3^2. a=2;b=3; p=2;q=2.
In fact, it is right:1+2+3+4+6+9+12+18+36=91
We know that a rational number is a number that is expressed in the form of p/q, where p and q are integers and q is not equal to zero. A fraction is also written in the form of p/q, where p and q are integers and q is not equal to zero.
If p is less than q, then p/q is called a proper fraction. For example, 1/2, 2/3, 3/4. The meaning of the fraction ‘1/2’ is 1 out of 2 equal parts. Similarly, 2/3 means 2 parts out of 3 equal parts. The numerator ‘p’ is the fraction of the denominator ‘q’, if you consider ‘q’ as one whole thing. Suppose you divide a circle into two equal parts, then that one part is equal to 1/2 of that circle. Similarly, if you divide the circle into three equal parts, then two of those three parts is equal to 2/3 of the circle.
If p is greater than q, then p/q is improper. Improper fractions are greater than 1 or less than -1. By using division algorithm, an improper fraction can be converted into a mixed fraction, which is sum of an integer and a proper fraction. For example, 7/4 is an improper fraction, which can be converted into 1¾. This is nothing but 1+¾. Similarly, -7/4=-1+(-¾)
Numbers of the form p/q can be represented as decimal numbers. For example, 5/2=2½=2.5; they are improper fraction, mixed fraction, and decimal fraction respectively. Similarly, 1/2=0.5; 1/4=0.25. Decimal numbers can be obtained by using division method.
When one or more digits of a decimal number recur in a fixed pattern they are called recurring fraction. For example, 1/3 = 0.3333333…..; 2/3=0.666666……; 10/9=1.11111….
A recurring fraction can be converted into a proper or improper fraction of the form p/q. Thus, 0.33333333…. can be converted into p/q form in the following method:
Let x=0.333333; multiply both sides by 10 because there is only one digit is recurring.
We get, 10*x=3.333333; now, (10*x-x)=3; 9*x=3; therefore, x=3/9 or 1/3. The logic is that we are converting the right hand side into a whole number.
How to find Least Common Multiple and Highest Common Factor
Generally, we confuse about what we have found: Is that LCM or HCF? A useful technique is remembering the two key terms ‘multiple’ and ‘factor’. Multiple is generally a greater number and factor is a smaller number. Suppose, you are asked to find LCM of two numbers, then your answer should be a number greater than or equal to the greater of the two numbers. Similarly, for HCF, your answer should be a number less than or equal to the smaller of the two numbers. For example, the LCM of 3 and 5 is 15, which is greater than 5; the HCF of 3 and 5 is 1, which is less than 3. At the same time, the LCM of 3 and 9 is 9, which equal to greater of 3 and 9; the HCF of 3 and 9 is 3, which is equal to smaller of 3 and 9. If you notice, in the first set, the two numbers are prime to each other; in the second set, there is a common factor.
When small numbers are given, you can actually calculate LCM and HCF in your mind. To find LCM, first consider the highest of the given numbers. In our example, in the first set, the numbers are 3 and 5. First, consider 5; can 3 evenly divide 5? Your answer is ‘no’. Now, consider the next multiple of 5, that is, 10; can 3 evenly divide 10? Your answer is ‘no’. Take the next multiple of 5, that is, 15; can 3 evenly divide 15? Your answer is ‘yes’. Therefore, the LCM of 3 and 5 is 15.
To find the HCF of 3 and 5, first, consider the smaller of the two, that is, 3; can 3 evenly divide 5? Your answer is ‘no’. Now, you have to consider a number smaller than 3 that can evenly divide both the given numbers 3 and 5. If you check you will know that only the number 1 can divide evenly both the numbers. Therefore, 1 is the HCF of 3 and 5.
When more than two numbers are given you can apply the same rules given above provided they are smaller. If the given numbers are greater, then you might take more time to calculate in your mind, though some students can do that.
Prime Factor Method
Consider the numbers 36 and 40.
36 = 2^2 * 3^2
40 = 2^3 *5
To find L.C.M. of the given numbers, consider all prime factors in their highest power/degree. They are 2^3, 3^2, and 5. When you find the product of these numbers, you get the L.C.M. of the two numbers. 8*9*5 = 360.
To find H.C.F. of the given numbers, consider only the common factors in their lowest power/degree. When you find the product of these numbers you get H.C.F. There is only one common factor in the given two numbers and it is 2. Its lowest power/degree is 2. Hence, the H.C.F. of the given two numbers is 4.
It’s useful to notice that we have considered all the factors in their powers/degrees while finding L.C.M. and H.C.F. It turns out that the product of the two numbers is equal to the product of their L.C.M. and H.C.F.
Product of two numbers = Product of their L.C.M. and H.C.F.