MHB How Can Rational and Irrational Numbers Combine in Arithmetic Operations?

[mathdad]

Give an example of irrational numbers a and b such that the indicated expression is (a) rational; (b) irrational.

1. a + b

2. a/b

Must I replace a and b with numbers that create a rational and irrational number?

 

[HallsofIvy]

Give an example of irrational numbers a and b such that the indicated expression is (a) rational; (b) irrational.

1. a + b

2. a/b

Must I replace a and b with numbers that create a rational and irrational number?

Your "location" is given as "NYC" so this shouldn't be a language difficulty?! The problem says "give an example of irrational numbers a and b". So yes, you are to replace a and b with irrational numbers. Since you have 2 problems and there are two questions for each, four different answers are required. You must give four a, b pairs of rational numbers. No complicated formulas are required, just some fundamental thinking.

 

[mathdad]

Question 1

a + b

Let a = pi, b = e

What about pi + e?

What about a = sqrt{2}, b = 4pi?

sqrt{2} + 4pi

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Question 2

What about sqrt{3}/pi?

What about sqrt{2}/sqrt{3}?

 

[MarkFL]

Question 1

a + b

Let a = pi, b = e

What about pi + e?

What about a = sqrt{2}, b = 4pi?

sqrt{2} + 4pi

Both of those sums would be irrational. (Yes)

Can you think of two irrational numbers whose sum is rational?

 

[mathdad]

Your "location" is given as "NYC" so this shouldn't be a language difficulty?! The problem says "give an example of irrational numbers a and b". So yes, you are to replace a and b with irrational numbers. Since you have 2 problems and there are two questions for each, four different answers are required. You must give four a, b pairs of rational numbers. No complicated formulas are required, just some fundamental thinking.

What does location have to do with my understanding of math? Why bring up language? There are questions that simply do not make sense as written. Why turn this into a race issue?

Help me understand the textbook or skip my questions. Who cares where I am from? By the way, I live in NYC but was born outside the USA. Now, what does this have to do with math? I am a foreigner with two CUNY degrees (NYC). I also know people that never set foot in college with a poor English background who easily understand mathematics.

You can see that I struggling with questions that others easily understand. You should be encouraging people not putting them down bringing up location and language. My questions are all related to math not English composition and grammar. Also, if you are going to continue answering my questions, break the material down for me to understand. To reply as written in math textbooks is a waste of time.

 

[mathdad]

Both of those sums would be irrational. (Yes)

Can you think of two irrational numbers whose sum is rational?

How about (10 + 2•sqrt{5}) and (5 - 2•sqrt{5})?

The sum of these two irrational numbers is

(10 + 2•sqrt{5}) + (5 - 2•sqrt{5}) =

10 + 5 which is 15 (a rational number). Yes?

Can you give me two other irrational numbers a/b that yield a rational answer?

 

[MarkFL]

How about (10 + 2•sqrt{5}) and (5 - 2•sqrt{5})?

The sum of these two irrational numbers is

(10 + 2•sqrt{5}) + (5 - 2•sqrt{5}) =

10 + 5 which is 15 (a rational number). Yes?

Yes, any pair of irrational numbers of the form:

$$a=\mathbb{Q}_1\pm\mathbb{I}$$

$$b=\mathbb{Q}_2\mp\mathbb{I}$$

will work, since the irrational quantity $\mathbb{I}$ will disappear when added.

Can you give me two other irrational numbers a/b that yield a rational answer?

What form would we need for the irrational part of the pair to be divided out?

 

[I like Serena]

For the record, we have to be a bit careful when adding irrational numbers. We cannot just conclude that the result will be irrational.
For instance it is not known whether pi + e is irrational or not. See here.
However, we can conclude that pi + pi = 2pi is irrational, since if it were not, dividing by 2 would prove that pi is rational, which is a contradiction.

 

[mathdad]

Yes, any pair of irrational numbers of the form:

$$a=\mathbb{Q}_1\pm\mathbb{I}$$

$$b=\mathbb{Q}_2\mp\mathbb{I}$$

will work, since the irrational quantity $\mathbb{I}$ will disappear when added.
What form would we need for the irrational part of the pair to be divided out?

How about (4 + pi)/(4 - pi)?

 

[MarkFL]

How about (4 + pi)/(4 - pi)?

Can that be rewritten as a rational number?

 

[mathdad]

Can that be rewritten as a rational number?

If I let pi be about 3.1, the answer is 1.675, a rational number.

 

[MarkFL]

If I let pi be about 3.1, the answer is 1.675, a rational number.

If you use a rational approximation for $\pi$, then you aren't actually using irrational numbers at all. :D

 

[mathdad]

Check this out:

(4 + pi)/(4 - pi) - Wolfram|Alpha Results

 

[MarkFL]

Check this out:

(4 + pi)/(4 - pi) - Wolfram|Alpha Results

A real transcendental number is also irrational...:D

 

[mathdad]

What about [2•sqrt{2}]/sqrt{2}?

This yields 2, a rational number.

 

[MarkFL]

What about [2•sqrt{2}]/sqrt{2}?

This yields 2, a rational number.

Yes, any pair of the form:

$$a=\mathbb{Q_1}\mathbb{I}$$

$$b=\mathbb{Q_2}\mathbb{I}$$

will work. :D

 

[mathdad]

Interesting question and topic. There is so much I don't know concerning the world of real numbers. Complex numbers is another interesting topic introduced in a later chapter by David Cohen in his fabulous Precalculus With Unit Circle Trigonometry 3rd Edition textbook. Most of my questions come from his book.