Real Numbers: Overview, Questions, Preparation

Parts of a number system

The number system is divided into two parts:

1. Imaginary numbers
2. Real numbers

Theorem 1: Euclid’s division lemma states that given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

Fundamental Theorem of Arithmetic

Theorem 2:  Fundamental theorem of arithmetic states that every composite number can be expressed ( factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Theorem 3: Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.

Theorem 4: Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form, p/q where p and q are coprime, and the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers.

Theorem 5: Let x =p/q be a rational number, such that the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

Theorem 6: Let x =p/q, where p and q are coprime, be a rational number, such that the prime factorisation of q is not of the form 2n5m, where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).

Importance and Weightage

The knowledge of real numbers is essential for your higher standards as they are the basics of mathematics.

• Prove that 3is irrational.

Let us assume that 3 is not irrational; that is, it is rational. This means that:

3 = a/b, where a and b are integers and b is not equal to 0.

Now, 

3 =a/b

b3 = a

Now, we square both the sides, we get

3b2= a2

From this, we can see that a2is divisible by 3, and a is also divisible by 3. 

Let's take another integer, say, c

As a is divisible by 3, we can say that

a=3c

Now,

3b2= 9c2

b2= 3c2

By this equation, we can see that b2 is divisible by 3, and thus b is also divisible by 3. 

Thus, a and b have a common factor 3.

Thus, 3 is irrational. 

• Given that HCF (306, 657) = 9, find LCM (306, 657).

We know that,

HCF * LCM = Product of two numbers

9* LCM = 306*657

LCM= (306*657)/9

LCM = 223338

• There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet again at the starting point?

We can find their meeting time by calculating the LCM of 18 and 12.

LCM(18,12) = 2*3*3*2*1 = 36

Therefore, Sonia and Ravi meet after 36 minutes at the starting point.

1.Real numbers are made up of what?

Real numbers constitute all rational and irrational numbers.

2.Is the negative of an irrational number also irrational?

Yes.

3.Is every irrational number a surd?

Every sued is an irrational number, but every irrational number is not a surd.

4.State any one identity of positive real numbers.

ab=a b

5.Is the prduct of a non-zero rational number and an irrational number rational or irrational?

Always irrational.