Q27. - Algebra 2

Question

Name the sets of numbers to which all of the following numbers belong. Then arrange the numbers in order from least to greatest.

$2.\overline{49}, 2.4\overline{9}, 2.4, 2.49, 2.\overline{9}$

Step-by-Step Solution

Answer

$2.\overline{49}, 2.4\overline{9}, 2.4, 2.49, 2.\overline{9}$ belongs to the sets of rational number and real number.

Order of numbers from least to greatest is $2.4, 2.49, 2.\overline{49}, 2.4\overline{9}, 2.\overline{9}$

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 - Find the sets to which given numbers belongs

Given numbers are in decimal form which are either terminating or repeating. So all numbers are rational number. Also, using above venn diagram, a rational number is a real number.

Thus, the numbers $2.\overline{49}, 2.4\overline{9}, 2.4, 2.49, 2.\overline{9}$ belongs to the sets of rational number and real number.

Step 4 - Arrange number from least to greatest

Write all numbers up to three places of decimal

$\begin{array}{c}2.\overline{49}=2.494\\ 2.4\overline{9}=2.499\\ 2.4=2.400\\ 2.49=2.490\\ 2.\overline{9}=2.999\end{array}$

• From first decimal place 9 is the largest number present in $2.\overline{9}$ so it is greatest number 
• From second decimal place 0 is the smallest number in $2.4$ so it is smallest number
• From third decimal place 0 is the smallest number in $2.49$ so it is the second smallest number.
• From third decimal place 4 is greater than 0 and smaller than 9 so $2.\overline{49}$   is the third smallest number.
• From third decimal place 9 is the larger number so $2.4\overline{9}$  is the fourth smallest number

Thus, order is from least to greatest number is $2.4<2.49<2.\overline{49}<2.4\overline{9}<2.\overline{9}$