Concepts Of Number System

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number-system

Number System Concepts with Flow Chart

The number system is the basic construct of mathematics. 1 and 2 questions on the number system are regularly asked in various competitive exams. Its knowledge helps to resolve different questions based on even/odd/whole/divisible/prime/co prime/rational/irrational/fractions/numbers and associated with divisibility, order, addition, subtraction, multiplication, ascending, descending, inverse numbers. These questions are solved in different methods. By practicing various types of problems we can achieve success in this chapter. let’s get into the chapter in this chapter we have Number System and Divisibility.

Number system Flow chart

Digits Used in Number System Concepts

The numbers 0,1,2,3,4,5,6,7,8 and 9 are called digits. Here the digit ‘1’ is noted as UNITY.

What are Natural Numbers?

All counting numbers are called Natural Numbers.

It is denoted by N. Where N={1,2,3,4,5…………..∝}.

What are Whole Numbers?

When zero is added to the set of natural numbers then they are referred to as Whole Numbers. It is denoted by W . Where W={0,1,2,3,4,5…………..∝}.

What are Even Numbers and Odd Numbers?

Even Numbers: 

The numbers which when divided by 2 gives reminder zero are Even numbers.

It is denoted by E . Where E={-∝…………-6,-4,-2,0,2,4,6…………..∝}.

Odd Numbers:

The numbers which when divided by 2 don’t give reminder zero are Even numbers.

It is denoted by O. Where O={-∝…………..-5,-3,-1,1,3,5…………..∝}.

Facts to Know on Even and Odd Numbers in Number system:

  1. sum or product of even numbers is even.
  2. The difference of even numbers is even.
  3. The product of a certain amount of numbers is even, then at least one of the numbers should be even.
  4. The product of a certain amount of number is odd, then none of them are even.i,e the product of any number of odd numbers is odd.
  5. The sum of odd numbers depends on the amount of numbers.
    a) Let the amount of numbers is even, then the sum is even.
    Ex: odd+odd=even            i,e 3+5=8.
    b) Let the amount of numbers is odd, then the sum is odd.
    Ex: odd+odd+odd=odd    i,e  3+5+1=9.

Integers:

The set of whole numbers natural numbers with the negative sign are included, then it becomes a set of Integers. It is denoted by I or Z. I={-∝…….-3,-2,-1,0,1,2,3……..∝}
  a) negative integers:{-∝…….-3,-2,-1 }
  b) positive integers:{0,1,2,3……..∝ }

What are Rational Numbers and Irrational Numbers?

Rational Numbers:

Any number that can be written as p/q form, where p,q are integers and q≠0 is a Rational Number. It is denoted by Q.
Ex: 3/4,-4/5, 0.9 …..

Irrational Numbers:

Non terminating and non-recurring decimal fractions are irrational numbers.
Ex: √3,2√5,e,∏=3.141592653  ….

Real Numbers:

The set of all the rational and irrational numbers together is called real numbers.

Cyclic Numbers:

Cyclic numbers are those numbers of n digits that once increased by the other number provide equivalent digits in a very totally different order.

Prime Numbers:

A natural number that does not have any factors other than unity and itself is called a prime number.
Ex:{2,3,5,7,11,13,17,19……}

Composite Numbers(Divisible Numbers) in Number System Concepts

A number that has factors as 1 and itself is named a composite number.
Ex: {4,6,8,9,10,12,14,16………}

How to Identifying a Prime Number?

Le ‘n’ be the number that is to be known as prime or not.
step1: Find the greatest integer (let it be k) less than or equal to the square root of n. i,e  K≤√n.
step2: Now, divide the number by all the prime numbers that are less than or equal to  K.
step3: If n is divisible by anyone of the prime numbers less than or equal to K then, n  is not a prime number.

Note:

  • 1 is considered as neither a prime nor a composite number.
  • 2 is the only even prime number. Remaining even numbers are composite.
  • Any prime number greater than 3 is often written in the form of 6m-1. Here m is a natural number.

Co-primes (Relative Primes):

Two numbers that have no common factor except 1 are called co-prime numbers.
Ex: (9,16) , (80,81),()

What are the Perfect Numbers?

Let the sum of all the factors of a number, including 1 and the number itself is double the number then, the number is called a perfect number.
Ex: let us consider the number 28 its factors are 1,2,4,7.14: Now their sum is: 1+2+4+7+14+28=56 which is double the number 28, therefore 28 is a perfect number.

Hierarchy of Arithmetic Operations in Number System Concepts:

The arithmetic operations have performed a rule referred to as BODMAS RULE. The arithmetic operations are to be carried out in the sequence of letters that appear in BODMAS.
It stands for:
B=Bracket
O=of
D=Division
M=Multiplication
A=Addition
S=subtraction.

B=Brackets has four types of brackets in number system:

                           a) Virnculum: This is represented by a bar  on the top of numbers
                           b) Simple bracket: represented by ()
                           c) Curly brackets: represented by {}
                           d) Square brackets: represented by [ ]

Here is an expression the brackets are to be opened in the order of vinculum, simple brackets, curly brackets, and square brackets.
i,e [ { ( – ) } ] brackets are to be opened from inside to outside.

Powers(involution):

Multiplying a number by itself is termed as raising the number to a power.
Ex: 5×5 = 5² .

Roots:

The root of a number(P) is the number formed which when raised to a certain power and is equal to number P.   
(√P)²=P. 

Complex Numbers:

Z=a+ib is termed as complex number, Here a and b are real numbers, b≠0 and i=√-1.

Additive Identity:

Let a+0=0 then, 0 is called additive identity.

Additive Inverse:

Let a+(-a)=0 then, a and -a are called additive inverse to each other.
Ex: 5+(-5)=0 then additive inverse of 5 is -5.

Multiplicative Identity:

If a×1=a then, 1 is called multiplicative identity.

Multiplicative Inverse:

If a×b=1 then, a and are said to be in multiplicative inverse to each other.
Ex: 4×¼=1. so 4 is the multiplicative inverse of 1/4.

Divisibility Rules in Number System Concepts

Though Divisibility questions are not asked directly, it’s knowledge is very essential to solve different questions in simplification.

  1. Divisibility by 2:  A number is divisible by 2 if it’s last digits are 0 or even number.
  2. Divisibility by 3: A number is divisible by 3 if the number formed by the sum of its digits is divisible by 3.
  3. Divisibility by 4: given number is divisible by 4 if the number formed by its last two digits is divisible by 4.
  4. Divisibility by 5: A number is divisible by 5 if it’s the last digit is 0 or 5.
  5. Divisibility by 6: A number is divisible by 6 if it is divisible by both the numbers 2 and 3.
  6.  Divisibility by 8: A number is divisible by 8 if the number formed by its last three digits is exactly divisible by 8.
  7. Divisibility by 9: A number is divisible by 9 if the number formed with sum of its digits is divisible by 9.
  8. Divisibility by 10: A number is divisible by 10 then its last digit should be 0.
  9. Divisibility by 11: A number is divisible by 11 if the difference of the sum of digits in odd places in the number and the sum of digits in the even places should be either equal to 0 or a multiple of 11.
  10. Divisibility by 12: A number is divisible by 12 if the number is divisible by both the numbers 3 and 4.
  11. Divisibility by 14: A number is divisible by 14 if it is divisible by both the numbers 2 and 7.
  12. Divisibility by 15: A number is divisible by 15 if it is divisible by both the numbers 3 and 5.
  13. Divisibility by 16: A number is divisible by 16 if the number formed by its last four digits is exactly divisible by 16.
  14. Divisibility by 18: A number is divisible by 18 if it is divisible by both the numbers 2 and 9.

Here for 7,13,17 and 19, the concept of osculators should be applied in Number System Concepts.

Divisibility Rule For 7: For a number to be  divisible by 7
a)  Take the last digit of the number given for divisibility and double it.
b) Subtract this number from the given original number.
c) If the resultant new number formed is either a 0 or is divisible by 7 then, the whole number is divisible by 7.
Ex: let us consider a number 161.
the last digit is 1, double of it 1×2=2: subtract from given number 16-2=14: 14 is divisible by 7 then 161 is also divisible by 7.

Divisibility by 13: For a number to be  divisible by 13
a) Take the last digit of the given number for divisibility and multiply it with 4.
b) Add this number with digits in the original number.
c) If the new number formed is divisible by 13 then, the whole number is divisible by 13.
Ex: let’s consider the number 156. 
last digit 6    multiply it with 4 6×4=24: add this result with given number 15+24=39: 39 is divisible by 13 then, 156 is also divisible by 13.

Divisibility by 17: For a number to be divisible by 17
a)  Take the last digit of the number for divisibility and multiply it with 5.
b) Subtract the result from the remainder of the digits within the original number.
c) If the resultant new number formed is 0 or it is divisible by 17 then, the whole number is divisible by 17.
Ex: let’s consider the number 272.
the last digit is 2 2×5=10: 27-10=17: 17 is divisible by 7 then 272 is also divisible by 17.

Divisibility by 19: For a number to be  divisible by 19
a) Take the last digit of the given number for divisibility and double it.
b) Add the resulting number to the rest of the digits in the original number.
c) If the resultant new number formed is divisible by 19 then, the whole number is divisible by 19.
Ex: let’s consider the number 323. 
the last digit is 3×2=6: 32+6=38: 38 is divisible by 19 then, 323 is also divisible by 19.

Read More: Number System Problems

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