Number System

Question 1

Euclid’s Division Lemma

Answer 1

Dividend = (Divisor x Quotient) + Reminder

Question 2

Fundamental Theorem Of Arithmetic

Answer 2

Every Composite Number can be written as the product of prime factors only.

Question 3

Proving that something is irrational

Answer 3

Proof: Let us assume that, sqrt(2) is rational
So, sqrt(2) = x/y (x,y are integers and y is not = 0)
Let x and y be co-primes.
2=x^2/y^2 (squaring on both sides)
2y^2=x^2
2 divides x^2
2 divides x
sore some integer z,
x=2z,
2y^2=4z^2
y^2=2z^2
2 divides y^2
2 divides y

This means that x and y have 2 as a common factor
But it contradicts the fact that x and y are co-primes.
Hence, sqrt(2) is irrational

Question 4

Expressing terminating rational decimals

Answer 4

p/q where (q is not = 0 ; q=2^m * 5^n)

Question 5

Expressing repeating rational decimals

Answer 5

p/q where (q is not = 0 ; q is not =2^m * 5^n)

Question 6

Number Systems

Answer 6

N = Anr^n +An-1r^n-1…. a1r+a0
(N= any integer, r= base of the system and a0, a1….an = the required digits)

Question 7

Decimal to Binary Conversions Steps:

Answer 7

Step 1: Divide the number that
needs to be converted
by 2.
Step 2 : Now divide the quotient
obtained in after step 1
and divide that by 2 ,
write the quotient and
the reminder.
Step 3 : Continue these steps
until we get zero as the
quotient.
Step 4 : Write the remainders
from the bottom to top.

Question 8

Binary Arithmetic

Answer 8

Addition:
Elementary rules:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 2
1 + 1 = 10 (1 will be taken as a
carry)
Subtraction :
Elementary rules:
0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 ( This works when 1
is borrowed from
the higher column
which turns to 2
in this column)