Factorial

The factorial of a positive integer is product of all positive integer less than or equal to 1.  It is denoted by n! where n is a positive integer.  For example,   

 4! = 4 x 3 x 2 x 1 =24 

An older notation for the factorial is .   The notation n! was introduced by Christian Kramp in 1808.

Mathematically, the formula for the factorial is as follows. If n is an integer greater than or equal to 1, then

 n ! = n .( n - 1).( n - 2).( n - 3) ... (3).(2).(1)

The value of 0! is 1. Though it seems to be absurd but for current purpose we just need to memorize it. It helps simplify a lot of equations.

Factorial is used in mathematics particularly in Combinations and Permutations, science and engineering. 

Factorial of fraction (0.8) and negative number (-4.35) is possible but it needs the help advance mathematics like Gamma Function.

 Returns of factorials:

 0!      1 
 1!      1
 2!      2
 3!      6
 4!      24
 5!      120
 6!      720
 7!      5040
 8!      40320
 9!      362880
10!      3628800

Complex Number

Complex number is a number that consists of real number and imaginary number. It is expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (Here, i² = −1). So complex numbers are combination of real and imaginary number. For example,

5 + i,    68 + 3i,     0.8 − 0.5i,     −2 + πi,     √2 + i/2

Above numbers are example of complex number. All the numbers are in the form of a+bi.

Complex number helps us to solve certain types of equations that is not possible by real numbers.
Suppose ,the equation

 x² + 7 = − 9

It has no real solution, since the square of a real number cannot be negative. With the help of complex number we can reach to a solution. We know that imaginary unit is i where i² = −1. So, the equation can be written as,

x² = (− 9). (− 7) 

or x² = − 16 

or x² = (−1).(16)

or x² = i² . 16 

or x = 4i

Properties of complex quantities:
i)  If  a+bi = 0 then a=0, b=0

ii) If  a+bi = c+di  then a=c and b=d
So if two complex numbers are equal then real and complex part of first number will be equal to real and complex part of second number.

iii) The result of addition and multiplication of the two conjugate complex quantities is always real.
The complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign. for example, the complex conjugate of  3 + 4i is 3 − 4i [From wikipedia]
 (a+bi) + (a-bi)= 2a      (a real number)
 (a+bi) . (a-bi)= a²+b²  (a real number)

iv) The result of addition, subtraction, multiplication and division of two complex numbers(not complex conjugate) will always be a complex number.

(a+bi) + (c+di)   = (a+c) + (b+d)i

(a+bi) − (c+di)  = (a−c) + (b−d)i

(a+bi) . (c+di)    = (ac-bd) + (bc+ad)i

(a+bi) / (c+di)    = (ac+bd/c²+d²) + (bc-ad/c²+d²)i

v) The root of a complex number is a complex one.

     ⁿ√a+bi = x 

 Cartesian Form of Complex Numbers

Complex number is used in various field of science,engineering as well as in mathematics to explain and understand scientific phenomena. In engineering, it is sometimes denoted by j. 

Imaginary Number

We know there are natural numbers, real numbers etc. Like them, there is a imaginary number. It is denoted by 'i'. How does it come?

We know that square root of a positive number is always positive. For example: 4²=16 and √16=4. Here 4 is positive number. So, it satisfies the equation X² =16.

So square of any positive or negative number is a positive number.

But look at the equation:     x²+4=0 that implies x² = − 4

What value of x will satisfy this equation? There is no real number whose square is negative and that is what we have been told that we can't take the square root of a negative number.

To solve this problem, an Imaginary Number 'i' was proposed( or invented, discovered whatever it is) where  

i² = -1

and 

i=√-1

So, we can write √−4 as  √−1.√4 = i2 and √−7 as  √−1.√7 = i7.

Imaginary number 'i'  has interesting property. Higher integral powers of 'i' cycles through i,−1,− i and 1. 

i²     =  −1
i³     =  i².i  =  −1· i  =  − i
i⁴     =  i³.i  =  − i· i  =  − i²  =  −(−1)  =  1

i⁵     =  i⁴.i  =  1· i  =  i

Similarly,
i^-1   =  1/i   =   i²/i   =  − i
i^-2   =  1/i²   =   1/-1   =  − 1
i^-3   =  1/i³   =   1/-i   =  (−1)(i^-1) = (−1)(− i) = i
i^-4   =  1/i⁴   =   1/i².i²   =  (−1)(−1) = 1
and
i⁰ =1

 

i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis
Photo courtesy: wikipedia

The concept of an imaginary number may be intuitively more difficult to grasp than that of a real number. It is widely used in mathematics and  physics. In electrical and electronic engineering, it is denoted by 'j' because 'i' is used as symbol of current.

Logarithm

Logarithm is closely connected to exponent. It is just inverse of exponent. If a^x= N then the Logarithm is:

x=log_aN

'x' is called logarithm of 'N' on base 'a'. This is stated as "log to base a of N equals x".

The base(here a) of the logarithm will always be a positive number and not equal to 1.

For example,

2³= 8     log₂8 = 3

3⁴= 81     log₃81 = 4

Logarithms can also be negative:

1/2¹= 2^-1     log₂1/2 = -1

1/5²= 5^-2     log₅1/25 = -2

But look at the example, log₄^(–16) = ?. What will be the logarithm?

If log₄^(–16) =x, then 4^x = -16.  What power of 'x' could possibly turn a positive 4 into a negative 16? This is impossible. So, in the exponential equation, a^x=N,  'N' will be any real positive number. We are raising a positive number to an exponent and so there is no way that the result can possibly be anything other than another positive number.

For any base a, log_aa = 1 and log_a1 = 0, since a¹ = a and a⁰ = 1.

Laws of Logarithm:

log_a(MxN) = log_aM + log_aN

log_a(M/N) = log_aM - log_aN

log_aM^P  = P log_aM

log_aP√M  = 1/P log_aM

There are 2 types of logarithms. The Common Logarithm and Natural Logarithm.

Logarithm which has base 10 is considered as Common Logarithm.  Henry Briggs first compiled 'Tables of Logarithms' in 1617. If base is not mentioned then it should be understood that base is 10. So log 4 means log₁₀4.

Before H. Briggs, John Napier, first introduced Logarithms in 1614 as a means to simplify calculations.The natural logarithm has the number 'e'  as its base. Here e ≈ 2.71828.

'e' is an irrational number. It is often called Euler's number. Napier used 'e' as base in Natural Logarithm. The natural logarithm of 'x' is generally written as ln x or log_ex.

Logarithm's main application was to reduce calculation or simplify calculation. Logarithm is widely used by scientists and engineers, and others to perform computations more easily. In Computer Science, binary logarithm(base 2 ) is used.

Progression

When in a sequence, a term is related to another term in mathematical way, then this sequence is called Progression. Progression is of 3 kinds: Arithmetic progression, Geometric progression, Harmonic progression.

Arithmetic Progression:

An arithmetic progression is a sequence of numbers where the difference of any two successive members is a constant. This constant could be positive or negative numbers. This constant is considered as 'common deference' of that sequence. Common difference is denoted by 'd'.

For instance, the sequence 1, 3, 5, 7, 9, 11, 13 … is an arithmetic progression with common difference of 2.

Similarly  50, 40, 30, 20, 10, 0, -10, -20.....  is an arithmetic progression with common difference -10.

To make sure whether a sequence or series is arithmetic or not, we just need to subtract each term from its successive term(except the first term). If the result of subtraction is same in all cases, then it is confirmed that the sequence or series is arithmetic.

If the initial term of an arithmetic progression is a₁ and the common difference of successive members is d, then the nth term of the sequence (a_n) is given by:

So, it advances in following way:

a, a+d, a+2d,  a+3d, a+4d. ..........

If d is the common difference and  we denote the n-th term of an arithmetic progression as a_n.  By S_n we denote the sum of the first n elements of an arithmetic series.

Arithmetic series means the sum of the elements of an arithmetic progression.

Then summation S_n can be obtained by :

Geometric Progression:

A geometric progression, is a sequence of numbers where each term (except the first one) is found by multiplying the previous one by a fixed number called the common ratio.

For instance, the sequence 3, 6, 12, 24, ... is a geometric progression with common ratio 2.

Similarly 1, 1/2, 1/4, 1/8 ... is a geometric sequence with common ratio 1/2.

In General we write a Geometric Sequence like this:

a, ar, ar², ar³, ...

where a is the first term and r ≠ 0 is the common ratio.

The n-th term of a geometric progression can be found by:   

a_n = ar^{n-1} 

where a is initial value and common ratio  is r.

A geometric series is the sum of the numbers in a geometric progression and it can be found by:

 

Harmonic Progression: 

Harmonic progression is obtained from arithmetic progression.  If a sequence is in arithmetic progression, then the sequence obtained by taking the reciprocal of every term in the sequence makes Harmonic Progression.

For example, if arithmetic progression is 3, 6, 9, 12 ..... then harmonic progression is

1/3, 1/6, 1/9, 1/12 ..........

Problems related to harmonic progression are generally solved by converting it into an arithmetic progression.

Relationship between Arithmetic, Harmonic and Geometric Progression:
Let a,b be two positive real numbers.
Then AM x HM=GM².

Sequences and Series

 

Sequence:  

When some numbers are written in some particular order, we can call it a sequence. For, example, take the numbers 

1, 3, 5, 7, 9, . . . . 

Here, we have a sequence of odd numbers. It is started with the number 1, which is an odd number, and then each successive number is obtained by adding 2 to give the next odd number. 

Here is another sequence: 

1, 4, 9, 16, 25, . . . .  

Above is the sequence of square numbers.   

20, 25, 30, 35, ...  

Here, each successive number is obtained by adding 5 to give the next number and sequence started with 20. 

Series:  

When we add up all the terms of a sequence, we will get series. For example, below is a sequence

1, 2, 3, 4   

The corresponding series is the sum 1 + 2 + 3 + 4, and the value of the series is 10.
Series is denoted by capital letter 'S'.  S 4 =1+4+9+16=30.
Or it can be showed by the Greek letter, called 'sigma':   

For example, 

Here n is lower index and telling us that from where the counting will be started and k is upper index which telling that where the sequence will be stopped.

Laws of Indices

If we multiply a number with this number, indices will be formed. Suppose 5x5x5=125=5³. Here we can call 5 as base and 3 as index. The index of a number shows how many times a base number  has been multiplied by itself. Index is also called Power. It is also known as Exponent. The plural of index is indices.

The main reason we use exponents is because it's a shorter way to write out big numbers. For example, let's say we want to express the following:

2×2×2×2×2×2 

we can see that 2 is multiplied by itself 6 times. This means we can write the same thing with 2 as the base and 6 as the exponent. That is 2⁶.

Law of Indices:  

To manipulate expressions involving indices we use rules known as the laws  of  indices.
When m and n are two positive integers --

1. a^m × aⁿ = a^{m + n}

2. a^m ÷ aⁿ = a^{m - n} (m ≥ n)

3. (a^m)ⁿ = a^{m × n}

4. a⁰= 1

5. a^-m= 1/a^m

Some extra rules:

6. (a × b)ⁿ = aⁿ × bⁿ 

7. (a/b)ⁿ = aⁿ ÷ bⁿ

8. (a)^1/n = ⁿ√a 

 
Last revised: 17/08/2016