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Explain that the sum, difference, and product of rational numbers is always a rational number

  • Last Updated : 20 Dec, 2021

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 a simple/easy system to indicate numbers and figures. It is a special way of showing numbers in mathematics and arithmetic forms.

Numbers

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. The numbers are divided into different sets in the number system based on the properties they reflect, for instance, all numbers generating from 0 and terminating at infinity are whole numbers, etc. Let’s learn about these numbers 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 to natural number but it also includes 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 characters involved in all the positive counting numerals, 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 contains a 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 the 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’.

Explain that the sum, difference, and product of rational numbers is always a rational number.

Answer: 

First, let’s know about Rational numbers 

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 form of integers, p/q form, most individuals 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.

Let’s consider two rational number as a/b and c/d,

Sum = a/b + c/d

Difference = a/b – c/d

product = a/b × c/d

These all are rational numbers because the numbers a, b, c and d are integers.

Let’s take an example to understand this problem,

a = 2, b = 1, c = 4, d = 1

Sum = a/b + c/d

= 2/1 + 4/1

= 6/1 (It is a rational number)

Difference = a/b – c/d

= 2/1 – 4/1

= -2/1 (Positive and negative do not effect rationality so, it is a rational number)

Product = a/b × c/d

= 2/1 × 4/1

= 8/1 (It is a rational number)

The sum and product of irrational numbers are not always irrational numbers.

For example: Consider two irrational numbers,

x = √3

y = 1/√3

So, the product of these numbers are

x(y) = √3 × (1/√3) = 1

Which is a rational number.

Similar Problems

Question 1: Is 0.924089924089924089924089924089… a rational number?

Solution:

The given number has a set of decimals 924089 which is repeated continuously.

0.924089  924089  924089 924089  924089

Same set is repeating.

Thus, it is a rational number.

Question 2: Sum of 0.2 + 4.2 is a rational number?

Solution:

First convert decimal into fraction form 0.2 = 2/10 and 4.2 = 42/10

Sum= 2/10 + 42/10

= 4.4 or 44/10 

 Thus, it is a rational number.

Question 3: A difference of 2.4 and 4.2 is a rational number?

Solution:

First convert decimal into  fraction form 2.4 = 24/10 and 4.2 = 42/10

Difference = 24/10 – 42/10

= -1.8 or -18/10

 Thus, it is a rational number.

Question 4: Product of 1.2 and 0.5 is a rational number?

Solution:

First convert decimal into fraction form 1.2 = 12/10 and 0.5 = 5/10

Product = 12/10 × 5/10

= 0.6 or 60/10

Thus, it is a rational number.

Question 5: Product of 1.5 and 3.2 is a rational number?

Solution:

First convert decimal into fraction form 1.5 = 15/10 and 3.2 = 32/10

Product = 15/10 × 32/10

= 4.8 or 48/10

Thus, it is a rational number.

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