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Special Cases of Numbers

Fibonacci Numbers: The number series in which the number is the sum of the previous two numbers are known as Fibonacci numbers. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

In general, for a Fibonacci number X, we can say that XI+2 = Xi+1 + Xi.

For any four consecutive Fibonacci numbers A, B, C, D, we always have: C2 – B2 = A * D For a Fibonacci series, The sum of the first n terms is Xn+2 – X2.

nth fibonacci number Fn is given by Perfect Numbers: If the sum of the divisors of a number excluding the number itself, is equal to the number itself, the number is called a

For example, the divisors of 6 are 1, 2 and 3. And 1 + 2 + 3 = 6, hence 6 is a perfect number.

Similarly, 28 and 496 are also perfect numbers.

Fn = [{(1 + 5)/2}n – {(1 – 5)/2}n] 1/5.

Cyclic numbers: A cyclic number is an integer of n digits, which when multiplied by any number from 1 to n, gives a product that contains the same digits.

Some important properties

1. Addition or subtraction of any two odd numbers will always result in an even number or zero. Addition or subtraction of any two even numbers will always result in an even number or zero.

2. Addition or subtraction of an odd number from an even number will result in an odd number. Addition or subtraction of an even number from an odd number will result in an odd number.

3. Multiplication of two odd numbers will result in an odd number.

4. Multiplication of two even numbers will result in an even number.

5.  Multiplication of an odd number by an even number or vice versa will result in an even number.

6. Division of an odd number by an even number will never be a whole number. The quotient can be either odd or even. But the remainder will always be odd.

7.          The quotient of dividing an odd number by another odd number can be odd or even.

8.          The quotient of dividing an even number by an odd number can be odd or even.

9.          The quotient of dividing an even number by another even number can be odd or even.

10.       An odd number raised to an odd or an even power is always odd.

11.       An even number raised to an odd or an even power is always even.

The Number Line

All numbers that can be expressed on a number line:

The number 0.3 will lie between 0 and 1, 2 is between 1 and 2, 7/3 is between 2 and 3. Hence, all numbers are real numbers.

The number on the left will be less than the number on the right. So –3 < –2, 1/2< 3/4 and so on.

To say that a number is between 1 and 4 means that it lies between 1 and 4 on the number line.

Mod values: The distance between a number and zero on the number line is called the mod or absolute value of the number. So |3| = 3 and |–4| = 4. Note that the absolute value is always positive.

Hence ⎜⎜a ⎜⎜ = + a, if a is positive or negative

Illustration 2:

What is the value of x in: ||x – 6 ||= 12?

Solution

There are two cases: (a) (x – 6) = 12 and (b) – (x – 6) = 12.
Solving both, we get x = 6 from the first and x = –6 from the second. The mistake that students normally do is that they forget the second case. The mod function must always be done by taking both cases.





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