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Closure Properties of Rational Number


Posted On : 2023-11-28 | Posted By : Admin

 



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The closure properties of rational numbers refer to certain properties that are preserved when performing certain operations on rational numbers. These closure properties demonstrate that certain operations involving rational numbers result in rational numbers. The closure properties for rational numbers are particularly relevant in mathematics, especially in algebraic structures.

Closure under Addition:

If you take any two rational numbers  
a/b and  c/d, where   b & d are not equal to zero
 then their sum 

(a/b)+(c/d) is also a rational number.
In other words, the sum of two rational numbers is always rational.

Closure under Subtraction:

If you take any two rational numbers 
a/b and c/d, where   b & d are not equal to zero
 then their difference 

(a/b)−(c/d) is also a rational number.
The difference of two rational numbers is always rational.

Closure under Multiplication:

If you take any two rational numbers  
a/b and c/d, where   b & d are not equal to zero
then their product 

(a/b)â‹…(c/d) is also a rational number.
The product of two rational numbers is always rational.

Closure under Division:

If you take any two rational numbers
  a/b and c/d, where   b & d are not equal to zero
 then their quotient  

(a/b)/(c/d) is also a rational number.

The quotient of two rational numbers (provided the denominator of the divisor is not zero) is always rational.

These closure properties demonstrate that the set of rational numbers is closed under the operations of addition, subtraction, multiplication, and division, as long as the denominators are not zero. These properties make rational numbers a field, which is a mathematical structure that satisfies certain algebraic properties. Fields, including the rational numbers, are fundamental in various branches of mathematics, such as algebra and analysis.




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