Page 35 - DCAP407_DATA_STRUCTURE

Data Structure

A logarithm is an exponent. The logarithmic function is defined as f(x)= log b x. Here, the base of the
algorithm is b. The two most common bases which we use are base 10 and base e

Consider the exponential equation 5 =25 where 5 is base and 2 is exponent.
2
The logarithmic form of this equation is:
log 525=2
Here, we can say that the logarithm of 25 to the base 5 is 2.
Factorial

The symbol of the factorial function is ‘!’. The factorial function multiplies a series of natural numbers
that are in descending order. The factorial of a positive integer n which is denoted by n! represents the
product of all the positive integers is less than or equal to n.
n!=n*(n-1)*(n-2)……2*1

5!=5*4*3*2*1=120

Fibonacci Numbers
In the Fibonacci sequence, after the first two numbers i.e. 0 and 1 in the sequence, each subsequent
number in the series is equal to the sum of the previous two numbers. The sequence is named after
Leonardo of Pisa, also known as Fibonacci.
Fibonacci numbers are the elements of Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765…….

Sometimes this sequence is given as 0, 1, 1, 2, 3, 5….. There are also other Fibonacci sequences which
start with the numbers:
3, 10, 13, 23, 36, 59…….
Fibonacci numbers are the example of patterns that have intrigued mathematicians through the ages. In
mathematical terms, the sequence F n of Fibonacci numbers is defined as:
F n = F n-1+ F n-2

Beginning with a single pair of rabbits, if every month each productive pair
bears a new pair, who become productive when they are 1 month old, how
many rabbits will there be after n months?
Assume that there are x pairs of rabbits after n months. The number of pairs
n
in n+1 month is x n+1 . Each pair produces a new pair every month but no
rabbit dies within that period. New pairs are only born to pairs which are at
least 1 month old, so there is an x new pair.
n-1
X n+1 = x + x
n
n-1
This equation shows the rules for generating the Fibonacci numbers.

Did you know? Fibonacci was the greatest mathematician of his age. He eliminated the use of complex
Roman numerals and made mathematics more accessible to the public by bringing the
Hindu-Arabic system (including zero) to Western Europe.
Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers. In the modular arithmetic, numbers wrap
around and reach a given fixed quantity, which is known as the modulus. This is 12 in the case of hours
and 60 in the case of minutes or seconds in a clock. In the 12 hour clock, the day is divided into two 12

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