Question

Name the sets of numbers to which each number belongs.

$\sqrt{10}$

Short answer

$\sqrt{10}$ belongs to the sets of irrational number and real number.

Step-by-step solution

- 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.

- 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

- Find the sets to which 10 belongs

As, $\sqrt{10}$ lies between whole number 3 and 4 which implies it is an irrational number.

So, using above venn diagram, $\sqrt{10}$ is an irrational number and a real number.

Thus, $\sqrt{10}$belongs to the sets of irrational number and real number.