Q61. - Algebra 2

Question

Determine whether each statement is true or false. If false, give a counterexample.

Every real number is irrational.

Step-by-Step Solution

Answer

Given statement is False.

Counterexample is $\frac{1}{2}$ as it is a real number but not a rational number.

Step 1 - Define different sets

• The set $\left\{1,2,3,4,5,...\right\}$ is called the set of natural number. 
• The set $\left\{0,1,2,3,4,5,...\right\}$ is called the set of whole number. 
• The set $\left\{-3,-2,-1,0,1,2,3,...\right\}$ is called the set of integer number.
• The set of rational number has numbers in the ratio form $\frac{m}{n}$, where m and n are integers and n is non-zero. The decimal form of a rational number is either a terminating or repeating decimal.
• A real number that is not rational is irrational. The decimal form of an irrational number is neither terminating nor rational.
• Combine sets of rational and irrational number is real number.

Step 2 - Relationship between different sets

Following venn diagram shows relationship between different sets

Where, the symbols denote 

N- Natural number

W- Whole number

Z- Integers

Q- Rational number

I- Irrational number

R- Real numbers

Step 3 - Check given statement is true or false.

Using image of step 2, set of real numbers does not lies inside the set of irrational numbers.

So given statement is false.

Example could be rational number which are not irrational number.

$\frac{1}{2}$ is a real number but not an irrational number.