Yes or No? If No, give a reason. Assume that a and b are...

Yes or No? If No, give a reason. Assume that a and b are nonzero real numbers.
(a) Is the sum of two rational numbers always a rational
number?
(b) Is the sum of two irrational numbers always an irrational
number?

Key Concepts

Counterexamples

A counterexample is a specific case which demonstrates that a general statement is false. In the context of the sum of two irrational numbers, providing a counterexample where the sum is rational serves as a concrete demonstration that the property does not hold universally.

Closure Properties Under Addition

The closure property under addition states that for a particular set of numbers, the sum of any two numbers from that set remains within the same set. While the set of rational numbers is closed under addition, meaning that the sum of any two rational numbers is always rational, the set of irrational numbers is not closed under addition, as the sum of two irrational numbers can sometimes be rational.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and denominator are integers, with the denominator not equal to zero. They have well-known closure properties under addition, subtraction, multiplication, and division (except by zero), which ensures that operations among rational numbers yield rational results.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction of two integers. Their decimal representations are non-repeating and non-terminating. They are significant in mathematics as they often arise in contexts like roots of non-perfect squares or transcendental numbers.

Transcript

00:03 Okay, we are talking about real numbers here.

00:07 We're specifically talking about rational versus irrational numbers.

00:10 We're going to start off by talking about the sum of two rational numbers.

00:14 A rational number is just anything that can be written as a fraction.

00:20 So if we're taking the sum of two rational numbers, we are just adding two fractions.

00:26 If the answer to this is also a fraction, then our answer will always be rational.

00:32 Let's think about how we add fractions.

00:35 In order to do that, we need a common denominator.

00:38 So the first fraction is missing a y, and the second fraction is missing a b.

00:44 And if we multiply, we end up getting a common denominator of by, which allows us to add these guys into just a single fraction.

01:04 And since our answer here can be written as a single fraction, that tells me my answer is going to be rational.

01:11 So the sum of two rational numbers is always rational.

01:23 Things get a little bit strange when we start talking about the sum of two irrational numbers.

01:28 Sometimes it seems very counterintuitive because it seems like if we add two irrational numbers, if we add pi plus pi, we're also going to end up with an irrational number.

01:39 But we are told that we have two irrational numbers, two numbers, a and b, that are irrational and not equal to zero.

01:52 So that means we can choose any value for a and b, as long as it meets these two criteria.

01:59 Well, i am going to choose a to be positive pi.

02:03 That's irrational...

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