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Is the quotient of two integers always a rational number?

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  • Last Updated : 21 Dec, 2021
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The number system involves dissimilar kinds of numbers such as prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form of facts as well as expressions suitably. For example, the integers like 20 and 25 shown in the form of figures can also be written as twenty and twenty-five. A number system or numeral system is defined as an simple/easy system to indicate numbers and figures. It is the special way of showing of numbers in mathematics and arithmetic form.

Number

Numbers are used in various arithmetic values appropriate to convey various arithmetic working like addition, subtraction, multiplication, etc., which are appropriate in daily lives for the cause of calculation. The worth of a number is determined by the digit, its place value in the number, and the stand of the number system. Numbers normally are also known as numerals are the numerical values used for counting, measurements, designating, and calculating elementary quantities. Numbers are the figures used for the cause of measuring or calculating numbers. It is constituted by numerals as 4, 5, 78, etc.

Types Of Numbers

There are different types of numbers. Numbers are distinguished among different sets in number systems based on the relationship they share and the characteristics they reflect. For instance, whole numbers generate from 0 and terminate at infinity. Lets learn about these types in more detail,

  • Natural numbers: Natural numbers are also known as positive numbers which count from 1 to infinity. The group of natural numbers is shown by ‘N’. It is the integer we normally use for counting. The group of natural numbers can be shown as N = 1, 2, 3, 4, 5, 6, 7,…
  • Whole numbers: Whole numbers are also known as positive numbers it is similar as natural number but it also include zero, which include 0 to infinity. Whole numbers do not contain fractions or decimals. The group of whole numbers is represented by ‘W’. The group can be shown as W = 0, 1, 2, 3, 4, 5,…
  • Integers: Integers are the group of character involve all the positive counting numeral, zero as well as all negative add up numerals which count from negative infinity to positive infinity. The group doesn’t involve fractions and decimals. The group of integers is expressed by ‘Z’. The group of integers can be shown as Z = …,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,…
  • Decimal numbers: Any integer value that contain decimal point is a decimal number. It can be represented as 2.5, 0.567, etc.
  • Real number: Real numbers are the group of integers that do not involve any imaginary value. It involves all the positive integers, negative integers, fractions, and decimal values. It is generally represented by ‘R’.
  • Complex number: Complex numbers are a group of numerals that involve imaginary numbers. It can be represented as x + y where “x” and “y” are real numbers. It is shown by ‘C’.
  • Rational numbers: Rational numbers are the numerals that can be represented as the ratio of two digits. It involves all the digits and can be represented in expression of fractions or decimals. It is represented by ‘Q’. It can be written in decimals and have endless non-repeating numbers after the decimal point. It is shown by ‘P’.

Is the quotient of two integers always rational numbers?

Answer: 

First, let’s learn about Rational numbers and Integers,

  • Rational number: Rational numbers are the divisor of two numbers in the form p/q, where p and q are numbers and q ≠ 0. Because of the basic formation of integers, p/q formation, most people find it hard to distinguish between fractions and rational numbers. When a rational number is divided, the output is in decimal form, which can be alternatively ending or repeating. 2,6,8, and so on are some examples of rational numbers as they can be shown in fraction form as 2/1, 6/1, and 8/1
  • Integers: Integers are the group of numerals involving all the positive add up numerals, zero as well as all negative add up numerals which include from negative infinity to positive infinity. The group doesn’t involve fractions and decimals. The group of numerals is shown by ‘Z’. The group of integers can be shown as Z = …,-8, -7, -6, -5, -1, 0, 1, 2, 3, 4, 5,… The integer with no decimal or fractional part from the group of negative and positive integers, including zero.
  1. Positive Integers: The integer numeral is positive if it is greater than zero. Example: 1, 2, 3, 4,…
  2. Negative Integers: The integer numeral is negative if it is less than zero. Example: -1, -2, -3, -4,… and here Zero is a whole number it is neither negative nor positive integer. It is a whole number. Z = {… -5, -4, -3, -2, -1, 0, …}

As per both the definition of Integer and Rational number, 

It is rightly said that the quotient of two number is always a rational number.

Step-by-step explanation:

  • If A and B are numbers it means that they are rational number and when quotient is caused by division of earlier two then the quotient must be rational as well.
  • With A is any number B is not being zero, then  A/B will always be rational number.
  • So, It is true that the quotient of two numbers is always a rational number.

Similar Problems

Question 1: Identify each of the following as irrational or rational: 1/2, 80/1244, 11, and √3.

Solution:

Since a rational number is the one that can be indicated as a ratio. This represent that it can be shown as a fraction wherein both denominator and numerator are whole numbers.

  1. 1/2 is a rational number as it can be shown as a fraction. 1/2 = 0.5
  2. Fraction 80/1244 is rational.
  3. 11, also be written as 11/1. Again a rational number.
  4. Value of √3 = 1.732050…. It is a infinite value and hence cannot be written as a fraction. It is an irrational number.

Question 2: Identify whether mixed fraction, 11/2 is a rational number.

Solution:  

The Simplest form of 11/2 is 3/2

Top one = 3, which is an number

Bottom one = 2, is an number and not equal to zero.

So, yes, 3/2 is a rational number.

Question 3: Identify whether the given numbers are rational or irrational.

  1. 1.75 
  2. 0.01   
  3. 0.5  
  4. 0.09   
  5. √3

Solution:

The given numbers are in decimal forms. To find whether the given integer is decimal or not, we have to convert it into the fraction form (i.e., p/q)

If the divisor of the fraction is not equal to zero, then the integer is rational, or else, it is irrational.

Decimal Number Fraction Rational Number
1.75 7/4 YES
0.01 1/100 YES
0.5 1/2 YES
0.09 1/11 YES 
√3   ? NO

Question 4: Is 0 a rational number?

Solution:

Yes, 0 is a rational number because it is an numeral, that can be written in any formation                                                       Such as 0/4, 0/5, where b i.e., (4, 5) is a non-zero numeral. 

It can be written in the formation: p/q = 0/1. Hence, we gather that 0 is a rational number.

Question 5: Identify a rational number between 4 and 5.

Solution:

Rational number between 4 and 5 = (4 + 5)/2

= 9/2

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