$\begingroup$ One last question to help my understanding: for a set of rational numbers, what would be its closure? Note : Addition of rational numbers is closure (the sum is also rational) commutative (a + b = b + a) and associative(a + (b + c)) = ((a + b) + c). This is called ‘Closure property of addition’ of rational numbers. Rational number 1 is the multiplicative identity for all rational numbers because on multiplying a rational number with 1, its value does not change. Closure depends on the ambient space. Proposition 5.18. In the real numbers, the closure of the rational numbers is the real numbers themselves. In mathematics, a rational number is a number such as -3/7 that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Therefore, 3/7 ÷ -5/4 i.e. Rational numbers can be represented on a number line. 0 is neither a positive nor a negative rational number. Example : 2/9 + 4/9 = 6/9 = 2/3 is a rational number. If a/b and c/d are any two rational numbers, then (a/b) + (c/d) is also a rational number. Every rational number can be represented on a number line. The algebraic closure of the field of rational numbers is the field of algebraic numbers. Subtraction Consider two rational number a/b, c/d then a/b÷c/d ≠ c/d÷a/b. Thus, Q is closed under addition. Additive inverse: The negative of a rational number is called additive inverse of the given number. Properties of Rational Numbers Closure property for the collection Q of rational numbers. Problem 2 : The sum of any two rational numbers is always a rational number. For example, the closure under subtraction of the set of natural numbers, viewed as a subset of the real numbers, is the set of integers. However often we add two points to the real numbers in order to talk about convergence of unbounded sequences. An important example is that of topological closure. The closure of a set also depends upon in which space we are taking the closure. Commutative Property of Division of Rational Numbers. Closure property for Addition: For any two rational numbers a and b, the sum a + b is also a rational number. number contains rational numbers. The reason is that $\Bbb R$ is homemorphic to $(-1,1)$ and the closure of $(-1,1)$ is $[-1,1]$. First suppose that Fis closed and (x n) is a convergent sequence of points x The notion of closure is generalized by Galois connection, and further by monads. Closure property with reference to Rational Numbers - definition Closure property states that if for any two numbers a and b, a ∗ b is also a rational number, then the set of rational numbers is closed under addition. Division of Rational Numbers isn’t commutative. -12/35 is also a Rational Number. A set FˆR is closed if and only if the limit of every convergent sequence in Fbelongs to F. Proof. Properties on Rational Numbers (i) Closure Property Rational numbers are closed under : Addition which is a rational number. 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