Why Babbage Is Called The Father Of Computer?

 

It isn't easy to select the boss of today's computer because there are tons of people who contributed in the field of computer science but Babbage is pioneer in that matter due to his invention and concept of the Analytical Engine.

Babbage, full name Charles Babbage, was a British mathematician.  In 1837 he came up with his concept for the first general mechanical computer,The Analytical Engine. It contained an Arithmetic Logic Unit (ALU), basic flow control, punch cards, and integrated memory. The design of this calculating machine strongly resembles the modern computer system and the overall organization is remarkably similar to the conceptual design of modern computers. Analytical Engine was intended to perform any arithmetical calculation using punched cards that would deliver the instructions, as well as a memory unit to store numbers. Ada Lovelace, another mathematician, who associated him by writing a program for the Analytical Engine.

Unfortunately, Babbage was not able to complete his project because of funding issues and thus his computer was never built during his lifetime. Babbage's son Henry Babbage constituted only a small part of the whole engine in 1910 which had some printing apparatus.

  

Though Babbage couldn't deliver the goods but he had that vision. Before Analytical Engine, he was involve in another project, the Difference Engine. The six-wheeled model was initially constructed and demonstrated to a number of audiences The difference engine was subsequently realized as actual working machines, so he was clearly on the right track. 

He is a pretty interesting person with some pretty interesting ideas, unfortunate in completing his project but no doubt, his idea was a leap in the field of computing. 

Functions and Equations

An equation is a statement where two expressions are equal. Here left and right side will be equal. 

Suppose x + 3 = 10. For a particular value x, the equation will be satisfied. If x=7 then 7+3=10. The left and right side are equal.

Function is a bit different. It is a relationship between two variables. Function is written as f(x)=x+3. It takes a set of value as input and return a particular output. The input variable f(x) is called Independent variable because it takes input as f(0),f(1),f(2) etc. and returns corresponding output as 3,4,5 etc. x + 3 is dependent on f(x) that's why x+3 is called as Dependent variable.

In equation, we need a particular value of variable to make it equal where function takes a set of  permissible input and returns regarding output.

What is Function in Mathematics?

A function in mathematics is just like relation between input and output. So a function takes input and returns an output. It is denoted as f.  Most of the time, you see it in below form.

f(x) = x² + 1 

For a given value of x, you will get the result of  x² + 1.

Value of x	Output
0 1 2 3 4	1 2 5 10 17

We see that for a particular value of variable x, we get a corresponding output. This phenomena is called function in mathematics.

We shall denote function of x as f(x), F(x), φ(x) etc.

The number x that we use for the input of the function is called the 'Argument' of the function. It may seems to us that we can pick any number as argument  and for this reason we can call it 'Independent Variable' and the output of the function, e.g. f(x), f(3), etc. depends upon the argument, can be called the 'Dependent  variable'.
We can draw a picture of a function on graph by taking argument-value pairs of the function and describing each by a point in the plane, with x coordinate given by the argument and y coordinate given by the value for that pair.

Word Problem of System of Equation

Q. Jimmy is 12 years older than Brandon. 17 years ago, Jimmy was 4 times as old as Brandon. How old is Brandon now?

Suppose Jimmy's current age is J and Brandon's age is B. Jimmy is 12 years older than Brandon then
J = B + 12
17 years ago Jimmy was 4 times as old as Brandon.
So, 17 years ago Jimmy's age was J-12 and Brandon's age was B-17 then
J-17 = 4(B - 17)
Put J = B + 12 in above equation.
B + 12 - 17 = 4( B - 17 )
B - 5 = 4B - 68
3B = 63
B = 21
Brandon is 21 years old.

Q. Your class has 40 students and some want to watch movie and some want to watch stage drama. The cost of movie ticket is 20$ and stage drama is 10$. Total cost of ticket is 500$. How many students went to watch movie and how many want to watch stage drama.

M= Number of student to watch movie
S = Number of student to watch stage drama

We know that total student is 40. Then
M+S = 40 
We also know total cost of ticket is 500$. Then
20M + 10S = 500
Put M = 40 - S  in above equation, it will become
20(40 - S) + 10S = 500
800 - 20S + 10S= 500
800 - 10S = 500
10S= 300
S= 30

Put this value in M+S = 40 then M + 30 = 40.
M = 10
30 students went to watch stage drama and 10 students went to watch movie.

Various Forms of Linear Equations(Two Variables)

There are three major forms of linear equations: point-slope form, slope-intercept form and standard form.

Slope-intercept form: 
y = mx + b
where m is slope and b is y-intercept.
Example: y = 2x + 1
Slope: m = 2
Intercept: b = 1

Point-slope form:
y − y₁ = m(x − x₁)
where m is slope and (x₁,y₁) is a point on the line.
Example: y - 1= 2 (x - 3)
Slope: m = 2
x₁ = 1
y₁ = 3

Standard form:
Ax + By = C
Where A,B,C are constants. 
Example: y + 3x= -10
A = 1
B =3
C = -10

Number of solutions to a system of equations

System of Linear Equations:
A system of linear equations or system of equations means two or more linear equations that are being solved simultaneously.

What is the solution of a system of equations?

When two equations meet or intersect at a point then this point is called solution of the system.

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line). This article reviews all three cases.

One solution:
This is the most common situation. Here, two lines meet at one point.
6x − 2y = 8​
4x + y = −1​​

Since the lines intersect exactly once, the system has exactly one solution.

No solution:
A system of linear equations has no solution when the graphs are parallel.
y = −x + 6​
5x + 5y = 15​​

We see two distinct parallel lines. We can confirm that the lines are indeed parallel, since the slopes of both lines are equal to −1. Since distinct parallel lines don't intersect, we conclude that the system has no solutions. 

Infinite solutions:
A system of linear equations has infinite solutions when the graphs are the exact same line.
​​​​​2x + y = 5​
14x + 7y = 35

The two lines are the same, they intersect infinitely many times. This means that the system has infinitely many solutions.

Solving Linear Equation by Substitution Method

The substitution method is another technique for solving systems of linear equations.

We're asked to solve this system of equations:
2y - 4x = 2
y -x = 4

Now we will find the value of x in second equation:
x = y - 4

Put the value of x in first equation.
2y - 4(y - 4) = 2
2y - 4y + 16 = 2
-2y = -16 +2
-2y = -14
y = 7

Now, put the value y=7 in second equation and we will get
7 - x = 4
x = 3


The solution of the linear system is (3, 7).

Solving Linear Equation by Elimination Method

 Elimination method is a way to manipulate systems of equations in order to solve them algebraically. This is actually very similar to the way we manipulate single equations in order to solve them.

The elimination method is a technique for solving systems of linear equations. Let's walk through a couple of examples.


We're asked to solve this system of equations:
3y + 11x = −10
4y − 11x​​​ = 17
We notice that the first equation has a 11x and the second equation has a −11x term. These terms will cancel if we add the equations together—that is, we'll eliminate the x terms:​​
3y + 11x = −10
4y − 11x​​​ = 17
-----------------
7y + 0 = 7

Solving for y, we get:
7y + 0 = 7
7y = 7
y=1
plugging this value back into our first equation, we solve for the other variable:
3y + 11x = −10
3.1 + 11x = -10
11x = -10-3
11x = -13
x = -13/11
The solution to the system is x=−13/11 and y=1

Not all time this system will work easily. Suppose a system like 4x +2y = 24 and −6x +2y = 4. If you add these two equations together, no variables are eliminated. This time we have to multiply any equation with -1.

4x   + 2y    =    24 

 − (− 6x + 2y ) = −(4)
or
4x   +  2y   = 24  

6x  −  2y    = −4
-----------------------
10x + 0y =20

x = 2

Put this value back into our first equation, we solve for the other variable:

4x + 2y =24

4.2 + 2y = 24

8 + 2y = 24 

2y = 24-8

2y = 16

y = 8

The solution to the system is x=2 and y=8

What is an equation?

What do you notice about each of the following?
11 + 3 = 15
25 - 5 = 20
The right hand side is equal to left hand side.
An equation is a statement that two expressions are equal. For example, the expression 12+3 is equal to the expression 16-1(because they both equal 15), so we can write the following equation:
12+3=16-1

When the left-hand side was equal to the expression on the right-hand side then the equation is called true equations. Let's make sure we understand what a true equation is.
8+2 = 5+5
7-3= 4+1
Here one thing must be clear to you that 5+3 is an expression and it is numerical expression but if the expression includes variable then it is algebraic equation. For example, the equation x+2=6 has a variable in it.

How can we solve algebraic equation?

3 + Z =  17

if Z=14, then 3 + 14 = 17.

How to Convert from Binary to Decimal

Suppose a binary number 1100₂  and we will convert it into decimal form. The binary number has base 2. In the number (1100)₂ , 0 is the least significant digit and 1 is most significant digit. We will multiply each digit with the base 2 to the power index number. So binary number (1100)₂ has following index number of each digit- 

3 2 1 0 ---- index number or order of digit. It starts from 0
1 1 0 0 ---- the number

So, the multiplication will be like:

1100  =  1×2³  +  1×2²  +  0×2¹  +  0 × 2⁰  = 8 + 4 + 0 + 0 = 12 

So,  the decimal value of binary number (1100)₂ is  (12)₁₀

The fraction number can also be done in similar way but that time the power will be negative. Suppose the number (0.011)₂ .

0.011 = 0×2^-1  +  1×2^-2  +  1×2^-3    = 0 + .25 + .125  = 0.375 

So,  the decimal value of binary number (0.011)₂ is  (0.375)₁₀

Writing Mathematical Expression

In mathematics, we can divide expression in two ways. That is -

1) Numerical Expression
2) Algebraic Expression

Here, by the word 'expression' or 'mathematical expression' we would like to define it as mathematical phrase that contains number,operator,variable or symbol etc.

Numerical Expression
A numerical expression contains numbers and operations. For example:

5+ 3
35÷7
53·4
94 - 31
102 - 9·3

All are example of numerical expression because it contains only number and operator.
Word problem: Robin plays football for 1.5 hours in the morning and 1 hour in the afternoon. He does this for a total of 30 days. Write a numerical expression with parentheses for the total number of hours Robin played football.

30 · (1.5 + 1)

Algebraic Expression
An algebraic expression contains numbers,operations and variables . For example:

5x + 3
9y - 8
6 + 3d
x/9

Word problem: Marco's mom gave him 'd' dollars for his allowance for a week. He also earned $14.55 for his newspaper route that week. How much money does Marco have? Write your answer as an expression.

d+14.55

Variable

Suppose you are going to buy a bicycle and you know it's price is 90$ and you also heard that there is a discount offer going on and you don't know what is the amount of discount. Then what will be the total price of that bicycle?

Total Price = 90$- Deduction 

So, if the discount is 10%(When you go to shop you see that 10% discount is going on) then the price is 

Total Price = 90$ - 10%

              = 90$ - 9$

       = 81$

So, the final price is 81$.

This unknown quantity of discount can be said as 'Variable' in mathematical context and a variable is just a symbol that can represent different values in an expression. So, in following expression Deduction is variable.

Total Price = 90$- Deduction 

It is representing different varying values, that's why called it variable.

In mathematics, variables are usually denoted by letters such as x,y,z,u,v etc.

So, the above expression can be written as 

Total Price = 90$ - X (Here X will be 10%)

Following are the example of variables:

x + 6

z + 10 

23y

x,z,y are example of variables. 

Continuous Variable:
If a variable x  can successively take all the values from a given number 'a' to another given number 'b', then x is called a continuous variable; otherwise, it is called a discrete variable.
The domain or interval of x in this case is denoted by a <= x <= b 

If a is omitted from the domain, it is indicated as a < x <= b  and it is said to be open at the left end.
The domain a <= x <= b is said to be closed and a < x < b is  open at both end.

Polynomial

A polynomial is a mathematical expression that have variables, coefficients and exponent. Polynomials are sums of these variables, coefficients and exponents. But polynomial does not contain a negative exponent.

An example of polynomial equation is:

x² - x - 1 = 0

In the word polynomial, poly  means 'many' and nomial means 'name' but here it will be used as term. So polynomial means "many terms". 

A polynomial in one variable with constant coefficients is given by

a_nxⁿ + a_n-1x^n-1 +.............. +a₂x² +a₁x +a₀

where a₀,  … ,a_n are constants and x is the variables.

Examples of polynomial:

• 6x 
• x - 4 
• 3x²  - 5x + 4
• -4y² - (³/₄)x 
• 2x⁵ – 5x³ – 10x + 9
• 6

6 could be considered as polynomial with 1 term.  Technically, the term polynomial should only refer to sums of many terms, but the term is used to refer to anything from one term to the sum of a zillion terms. However, the shorter polynomials do have their own names.

A one-term polynomial, such as 5x or 16x², may also be called a "monomial".

A two-term polynomial, such as 3x + y or x² – 4, may also be called a "binomial".

A three-term polynomial, such as 2a + b + c or 3x + 5y² – 3, may also be called a "trinomial".

In the polynomial expression, there should not be any negative exponent  (i.e x ^–2), the variable cannot be in the denominator (i.e 1/x), or any square root of variable. That's why below expression cannot be considered as polynomials:

4xy ^–2  -  because dividing by a variable is not allowed
3/(x+3) -  because dividing by a variable is not allowed
1/x - same as above. Variable is in denominator
√x - because square root is not allowed

Degree of terms:

The degree of a term is the sum of the exponents of all the variables in that term.

3x²  - 5x + 4

The above expression has 3 terms. First term has degree 2, second term has degree 1 and third term has no degree or zero degree or can be said constant term.

3yx² - 2x

Here, the first term has two variables and its degree is 3 because variable 'y' has power 1 and variable 'x' has power 2. So, 1+2=3. The second term has degree 1.

Naming of degree:

Degree	Name	Example
0	Constant	7
1	Linear	x + 7
2	Quadratic	x2− x + 2
3	Cubic	x3 − x2 + 5
4	Quartic	6x4 − x3 + x − 2
5	Quintic	x5− 3x3 + x2 + 8

This naming table is collected from Math is Fun.

The general form of a polynomial shows the terms of all possible degree.  Here, for example, is the general form of a polynomial of the third degree:  

3x³ + 2x² + 5x + 8

An equation that is in polynomial form is called Polynomial Equation.  Polynomials is used widely in the areas of mathematics and science.

Linear Equation

A Linear Equation is a mathematical expression.  It is an algebraic equation where one variable depends on other. It is generally appears in below form:

y = 6x + 2 

The above equation is in the form of y = mx + b, where m and b are constant.

It is an equation that forms a straight line on a graph. In this particular equation, the constant m determines the slope or gradient of that line, and the constant term b determines the point at which the line crosses the y-axis, that is also known as the y-intercept. So, in the equation y = 6x + 2, slope m is 6 and y-intercept b is 2.

Linear equation is simple in manner and  each term in linear equation will not have 

• Exponents (or powers). For example, x²
• Multiplication or division of  each other. For example: "x" times "y" or xy; "x" divided by "y" or x/y
• A root sign or square root sign (sqrt). For example: √x or the "square root x"; sqrt (x) 

The following equations are  all linear equations:

i) y = 2x+1             ii)  9x = 6+3y               iii) y/4 = 9 - x

Linear equation will be in other form also. For example

i) y = 4                ii) 8x + 4 = 0                iii) y - 3 = ¼(x - 2)

The third example is in point-slope form [ y-y1 = m(x-x1) ]

 

Linear equation graph

(Photo courtesy: hotmath.com)

Solving a linear equation:

i)           4x + 8 = 20
             4x = 12
             x = 12/4
             x = 3

ii)        ( (x-2)/3 ) + 1 =  2x/7
           (x + 1)/3  =  2x/7
           7 (x + 1)  =  6x
           7x +7  =  6x
           7x - 6x  =  - 7
            x  =  - 7

What is Coefficient in mathematics?

The coefficients are the numbers by which another number or symbol is multiplied. In a mathematical equation, a coefficient is a constant by which a variable is multiplied. Consider the following equations:

 7x² + 6x + 8

Here, x is a variable, and "7" and "6" are coefficients of x² and x respectively. Now consider another expression,

 7a² - 6xy + y + 8

Here is total 4 terms. In first term 7a²,  7 is coefficients of a². In the second term -6xy,  -6 is coefficients of xy and the third term y has no written coefficient but it is considered that it has 1 as coefficient. The last term 8  is a constant.

Take a look at following equation:

ax ³ + by ² = z

Here, a is coefficient of x ³ and b is coefficient of  y ². Variables are generally declared by the letter x,y,z etc.

Euler Number

Euler's number e is a mathematical constant and it is named after the Swiss mathematician Leonhard Euler. The pronunciation of Euler is 'Oiler' in English. The number e is also known as Napier's constant because the number  appeared on John Napier's work on logarithms that was published in 1618.   Leonhard Euler was not inventor of that constant but he introduced the letter e as the base for natural logarithms and Euler's choice of the symbol e is said to have been retained in his honor. Euler probably used this symbol in 1927.

The number e is a famous irrational number, and is one of the most important numbers in mathematics. It is also called as Transcendental number. The numerical value of e is :

2.71828182845904523536 (and more ...)

There are many ways of calculating the value of e.  

n	(1 + 1/n)n
1	2.00000
2	2.25000
5	2.48832
10	2.59374
100	2.70481
1,000	2.71692
10,000	2.71815
100,000	2.71827

Its value is approximately 2.718281828459045... and has been calculated to 869,894,101 decimal places by Sebastian Wedeniwski.

Another way we can calculate the value of e and it is below:

e=1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + ...

Here  "!" means factorial.

The first few terms add up to: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 +...... = 2.718055556

The sum of the values is 2.7182818284590452353602875 which is "e."

Permutation and Combination

Suppose there are 3 balls: Red, Green and Blue. In shortcut we can identify them as R,G and B. Now, how many was we can arrange these balls? Definitely in six ways.

RGB, RBG, GRB,GBR, BRG, BGR

Six ways. It can be found by 3! which is 3 x 2 x  1= 6.

But if we are told to arrange these 3 balls in such a way that we will pick only two at a time, then how many ways we can arrange or sort it. Let's see,

RG, RB, GB, GR, BG, BR 

This is also in six ways and this arrangement or sorting is called Permutations.

Let's think about this 3 balls again. How many ways we can pick 2 balls by ignoring their orders. 

RG, RB, BG 

or

GR, BR, GB

Only 3 ways. Both time we ignored their internal ordering or arrangement because RG and GR is same, only difference is order. So, the number of ways we picked 2 balls from 3 balls is called Combination.

Permutations:

The ordered arrangements of a set of distinct objects is called Permutations. This ordered arrangement could be of some elements. So, an ordered arrangement of r elements from n distinct object is called  r-permutation. It is denoted as ⁿP_r, nPr or P(n,r).

The number of r permutation of a set with n distinct element is

P(n,r)= n.(n-1).(n-2).....(n-r+1)

The first element of the permutation can be chosen in n ways because there are n elements at first. After that there will be (n-1) element will be left so, the second permutation can be done n-1 ways. Similarly, the third will be n-2 ways and so on. The permutation of rth element can be done in (n-r+1) ways.

P(n,r) = n.(n-1).(n-2).....(n-r+1)

Because we know that  n! =  n.(n-1).(n-2).....(n-r+1). (n-r)!

Example: 
How many different ways there are to select 3 winners(1st,2nd,3rd) from 100 different people?
Selecting 3 winners(1st,2nd,3rd) from 100 different people means selecting 3 ordered elements from a set of 100 different elements. Here 3 permutation of a set of 100 elements is P(100,3) = 100 x 99 x 98 = 970200

Combinations:
Unordered selection of some element from n distinct object is called Combinations. So, selecting r element from n distinct object by ignoring their order(no repetition) is called r combination. It is denoted as ⁿC_r, nCr or C(n,r).

The number of r combinations of a set with n elements, where n is a non-negative integer and r is an integer with 0 <= r <= n, equals

 

 

The formula can also be re-written as follows:

Example: 
How many ways are there to select 5 players from a 10 member tennis team?
The answer can given by the number of 5 combinations of a set with 10 elements. So the combination is:

C(10,5) = 10! / 5!5! = 252

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

Number Types

There are different types of number in mathematics. Different types of numbers have many different uses. We can also classified them as sets.

Natural Number:

The first type of number is 'natural' numbers. It is also called counting number. These are:

 1, 2, 3, 4, 5, 6, ... 40,41 ...100,101 ......
The set of natural numbers, denoted N, can be defined in N ={1,2,3,4}

Whole Number:

Same as Natural Number. Here it is together with zero:

    0, 1, 2, 3, 4, 5, 6, ...

So if we include 0 with natural number then it is whole number.

Integers:

Then come the "integers", which are set of zero, the natural numbers, and the negatives of the naturals. It is written without fractional part. These are:

    ..., –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, ... 

So +8 or -8 both are integer but 8.23 is not integer. 8.23 is a fraction. The set of integers, denoted Z, is formally defined as follows: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Rational numbers: 

The rational number is ratio of two integer numbers. So it can be written in the p/q form, where p and q are integers and q is not equal to zero. By dividing one integer by another integer may form a fraction.  For example, 5/2=2.5 but 4/2=2 does not make fraction.

So fractions are rational number but remember that fractions should be terminated (ending) or repeating decimals. Such as 1/2=0.5(fraction is terminated), 1.58/2= 0.79(fraction is terminated), or 4/3=1.333333.....(Here fraction is repeating). All these numbers are example of fraction. 

Irrational numbers:

Irrational means not Rational. Some numbers cannot be written as a ratio of two integers and they are called irrational number. For example:

√2=1.4142135623730950....... (fraction is non-terminating) 

π = 3.1415926535897932384626433832795 (and more...) 

Here fraction is endless. The decimal expansion of an irrational number continues without repeating.

Real numbers:
Real number includes all the number: natural,integer,rational and irrational numbers. It is denoted by R.

How to Convert from Decimal to Binary

The decimal number system has base 10 and binary number system has base 2. In study of computer science, we often need to convert a decimal number to a binary one. The following approach will teach how to convert a decimal number to a binary one.

The approach is simple. Take a decimal number 9 and now we will convert it to decimal.

Divide this number by 2 as it is the base of the binary numeral system. The remainder will always either be 0 or 1. Then keep this remainder and divide the quotient by 2. We will do this method until we reach the quotient as 0.

9 ÷ 2 = quotient 4 and remainder 1

4 ÷ 2 = quotient 2 and remainder 0

2 ÷ 2 = quotient 1 and remainder 0

1 ÷ 2 = quotient 0 and remainder 1  

Now start with the bottom remainder, read the sequence of remainders upwards to the top, we will get the binary number 1001. Here the last or bottom remainder is 1 as 1 ÷ 2 = quotient 0 and remainder 1 and top remainder is also 1 as 9 ÷ 2 = quotient 4 and remainder 1.

Take another decimal number 47 and its binary value is 101111. We will find it out now.

47 ÷ 2 =  quotient 23 and remainder  1

23 ÷ 2 =  quotient 11 and remainder  1 

11 ÷ 2 =  quotient 5 and remainder    1

5 ÷ 2  =  quotient 2 and remainder     1

2 ÷ 2  =  quotient 1 and remainder     0

1 ÷ 2 =   quotient 0 and remainder     1  

If start with the bottom remainder, read the sequence of remainders upwards to the top, we will get the binary number 101111.

Above we  have discussed about converting integer decimal into binary. What will we do in case of fractional number?  Suppose the number is .25 and how can we convert it into binary. We will do it in the following way:
First, we will multiply the fractional part by 2 and then keep the whole number part or the result. We will continue this approach until we reach to a full integer. So,
0.25  × 2 = 0.50  keep  0  (Here 0 is whole number part and .50 is fractional part)
0.50  × 2 = 1.0    keep  1  (Here 1 is whole number part and .0 is fractional part)
Now we will count it from top to bottom and we will get 01 as 0 is the top digit and 1 is the last digit.

So the binary of .25 is  0.01

Take another number  0.5625 and find its binary value.

0.5625  × 2 = 1.125  keep  1  (Here 1 is whole number part and .125 is fractional part)
0.125    × 2 = 0.25    keep  0
0.25      × 2 = 0.50    keep  0 
0.50      × 2 = 1.0      keep  1  

Start count from top to bottom  the binary number will be 0.1001

Binary Number

In numeral system, Binary Number System is simplest. The base or the radix of the binary system is 2. So, there are two digits: 0 and 1. Gottfried Wilhelm Leibniz, the co-inventor of Calculus, first talked about binary number in the paper 'Essay d'une nouvelle science des nombres' in 1701.

As there are two symbols: 0 and 1, so , all the numbers are represented with this two symbols. Below table shows how decimal numbers are represented in binary system -

Binary	0	1	10	11	100	101	110	111	1000	1001	1010
Decimal	0	1	2	3	4	5	6	7	8	9	10

Suppose a binary number 1010 that is equivalent to 10 in decimal. The rightmost bit is called  Least Significant Bit (LSB) and the leftmost bit is called Most Significant Bit (MSB). Then 1 is MSB and 0 is LSB.

Fractions can also be represented in binary system. Suppose decimal 22.75 can be written in binary as (10110.11). Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals. There is a method of converting binary number in other number system.

The binary system is used internally by almost all modern computers and computer-based devices. Each digit is referred to as a bit. Computer memory comprises small elements that may only be in two states - off/on - that are associated with digits 0 and 1. Such an element is said to represent one bit - binary digit. 

Binary arithmetic

Addition:
Addition is done exactly like adding decimal numbers. Below is just rule in binary addition like decimal system. When 1 is added with 1, the result will be 0 and there will be an extra bit 1 that is called Carry. This carry will have to be added in the next column.

0+0 = 0
1+0 = 1
0+1 = 1
1+1 = 0, and you carry a 1. 
The two binary numbers (110)₂=decimal 6 and (111)₂=decimal 7 can be add in the following manner. The answer will be (1101)₂ =decimal 13.
  110

+111
------------
 1101 

The above addition is for two positive numbers. But if it is the combination of positive and negative numbers then we have take other method. For example, when you handwrite a number that represent some physical quantity such as temperature, you can simply put a +  or - sign in front of the number to indicate that the number as positive or negative. Storing this value in computer is problematic.  To resolve this problem, it was decided that the most significant bit (MSB) will be reserved as sign data or bit.

To make computation easier with signed numbers, the magnitude of negative number is represented in a special form called 2's complement. It is formed by inverting each bit or digit and adding 1 to the result. Suppose in a 8 bit representation, the decimal number +5 can be written as 00000101 and decimal number -5 can be written as 10000101. Here we, at first, inverted 00000101(+5) and found 11111010. Then we add 1 with 11111010 and found 11111011(-5).  So addition of signed binary numbers can be done in following way -
0 0 0 0 1 1 0 1   (+13)
1 1 1 1 0 1 1 1   (-9) ( in 2's complement form )
----------------
0 0 0 0 0 1 0 0   (+4)

Subtraction:

Subtraction is same as decimal subtraction. When we subtract 1 from 0, the result will be 1 and we will hire 1 from next column. This hiring is called borrowing. Then the value of the next column becomes 0.

0 − 0 = 0
0 − 1 = 1, borrow 1
1 − 0 = 1
1 − 1 = 0
Here we subtract binary (1011)₂=decimal 11 from binary (10101)₂=decimal 21 and the result will be (1010)₂=decimal 10. In that subtraction method, larger number should be top and smaller number should be in bottom.  
 1 0 1 0 1
-  1 0 1 1
 ----------
    1 0 1 0

Multiplication:
Multiplication in binary follows same rule that we use in decimal.
0 x 0 = 0
1 x 0 = 0
0 x 1 = 0
1 x 1 = 1, no carry and no borrow

Here, we will multiply binary (110)₂=decimal 5 and (11)₂=decimal 3. The result will be (10010)₂=decimal 18.

          1 1 0
          x  1 1
     -------------
         1 1  0
      1 1 0
-- -------------

  1  0  0 1  0