Number System -1

Introduction

A number is a mathematical object used to count, measure, and label.

number

Above cartoon represents 1 to 9 but it does not represent 0. I think 0 is the most powerful number and bizarre number.If you put 0 after any number it will change the value of number by 10 though it does not contain any weight.To represent a number we need a number system, the most commonly used system of numerals is the Hindu–Arabic numeral system. It is also pronounced as Decimal System.Decimal system uses 10 digits that’s why it is called decimal system.If a number system uses 2 digits (0, 1) to represent a number then it is called binary number system

Number in English Binary Representation Decimal Representation
ZERO 0 0
ONE 1 1
TWO 10 2
THREE 11 3
FOUR 100 4
FIVE 101 5
SIX 110 6
SEVEN 111 7
EIGHT 1000 8
NINE 1001 9

Numerical Operations

There are 4 operations which can be performed on numbers.

  • Addition
  • Subtraction
  • Multiplication
  • Division

Out of these 4 operations, addition and subtraction are the most fundamental operations. We can do multiplication and division with the help of addition and subtraction as we know multiplication is repeated addition and division is repeated subtraction.

An Algorithm for you

Step 1: Take 2 positive integers (a and b)

Step 2: if a is equal to b then go step no 5

Step 3: if a > b then a= a-b or b=b-a

Step 4: Go to step 2:

Step 5: Print the value of a

What is the output of this operation? Try this operation on  following set of numbers

  • 5,10
  • 3,5
  • 9,12

For ( 5, 10 ), the output is 5

For (3, 5) the output is 3

For (9, 12) the output is 3

So the outcome is HCF of 2 numbers.

Look carefully, we are doing repeated subtractions, which is division. We are dividing 2 numbers. Let us perform step 1 to step 5

Step 1: Take 2 positive integers (a and b)

Step 2: if a is equal to b then go step no 5

Step 3: if a > b then a= a-b or b=b-a

Step 4: Go to step 2:

Step 5: Print the value of a

Step 1: Take 2 positive integers (a and b) a=5, b=10
Step 2: if a is equal to b then go step no 5 Here a is not equal to b
Step 3: if a > b then a= a-b or b=b-a a is not greater then b so b= 10-5 = 5
Step 4: Go to step 2: Now go to step 2:

Not at step 2: Both are equal.

Step 5: Print the value of a The outcome is 5.

Representation

In the universe, based on our understanding, every number can be represented in the form of complex numbers.

Representation of Complex Number: – A complex number has 2 parts – Real Part and Imaginary Part.

  • A complex numbers becomes real number, if imaginary part is zero.
  • A complex number becomes imaginary number, if real part is zero.

Z = a+ ib  (Z is a complex number, a is real part of it, ib is imaginary part of it, where a and b are real numbers)

i is an imaginary unit such that i2= -1

Complex numbers are used in

  • Electricity
  • Fractal Generation
  • Study of Fluid

(x +1)2 = -9

(x +1)2 = 9i2 (We know that i2= -1)

(x +1)2 = (3i)2

(x +1) = +3i or (x +1) = -3i

Hence x = -1 +3i or -1 -3i

Types of Number

number_2

 Number Equations

  • Complex Number = Real Numbers + Imaginary Numbers
  • Real Numbers = Rational Numbers + Irrational Numbers
  • Rational Numbers = Integers + Fraction
  • Integer = Whole numbers + the negative of the whole numbers
  • Whole Numbers = Natural Numbers + 0

Here, + means, need to include

Complex Number (C) x+iy
Real Numbers (R) 3, √2, 3/4
Rational Number (Q) Represented in the form of a/n
Integer (Z) …..-4,-3,-2,1,0,1,2,3,4……
Whole Number (W) 0,1,2,3,4….
Natural Number (N) Starts with 1

We can perform basic mathematical operations (Addition, Subtraction, Multiplication and division) on real numbers.

number3

Observation #1:
Notice that √(9) is a natural number. It is because √(9) = 3
Observation #2:
Notice that the only difference between natural numbers and whole numbers is the zero.
Whole numbers = Natural numbers + zero
Observation #3:
Notice that the difference between whole numbers and integers are the negative numbers.
Integers = Whole numbers + the negative of the whole numbers
Observation #4:
All integers are fractions. Not all fractions are integers
Example: -2 is an integer and can be written as -2/1 to make it a fraction.
However, -1/3 = -0.333333333 is not an integer
Observation #5:

Fractions can be written as a terminating decimal or a repeating decimal

Example: 1/2 = 0.5 and 0.5 is a terminating decimal. 1/3 = 0.3333333 and 0.3333333 is a repeating decimal

Observation #6:
Rational numbers = Integers + fractions
Observation #7:
Irrational numbers are numbers that cannot be written as a fraction
Example: pi= 3.14…, 2.224879566117426874, √(7)
Another way to see them is that they are neither repeating decimals nor terminating decimals
Observation #8:
Real numbers = rational numbers + irrational numbers

Problems

Q1. Add

1+2+3+4+…………………………………………………….+100

Solution :-

Sum of n consecutive numbers = n*(n+1)/2

Here n = 100

Sum = 100* 101/2 = 5050

We can also solve this problem without applying any formula

0 1 2 3 4 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 98 99 100
100 99 98 97 96 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 1 0

Sum of each column is 100, So 100 is repeating 101 times. That means it becomes 100*101 = 10100

Now we have to divide 10100 by 2 to get the answer so answer is 10100/2 =5050

Think, why we need to divide 10100 by 2.

Q2. Solve 1/ 6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 +1/90

Solution :

1/ 6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 +1/90

It can be written as

1/2.3 + 1/3.4 + 1/4.5 + 1/5.6 + 1/6.7 + 1/7.8 + 1/8.9 + 1/9.10

You can write

1/2.3 = 1/2 – 1/3

1/3.4 = 1/3 – 1/4

1/4.5 = 1/4 – 1/5

1/5.6 = 1/5 – 1/6

1/6.7= 1/6 – 1/7

1/7.8= 1/7 – 1/8

1/8.9= 1/8 – 1/9

1/9.10 =1/9 – 1/10

Now adding all of them,

1/2.3 + 1/3.4  +1/4.5 + 1/5.6 +1/6.7 +1/7.8 +1/8.9 +1/9.10

=  1/2 – 1/3 +  1/3 – 1/4 +1/4 – 1/5  + 1/5 – 1/6 + 1/6 – 1/7  + 1/7 – 1/8 + 1/8 – 1/9  + 1/9 – 1/10

= 1/2 – 1/10

= 4/10

=2/5

Q 3. Prove mn = 0

m can be written as  = mn-n

We know that, ma-b = ma/mb

So, mn-n = mn/mn = 1

Note :- Some information and pictures are taken from Internet. I don’t remember exact source but would like give credit to them.

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