Irrational Numbers a and b used in various expressions

[nycmathguy]

Homework Statement

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

Relevant Equations

None.

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

1. a +b

2. a•b

3. a/b

4. a - b

What exactly is this question asking for? Can someone rephrase the statement above?

Thanks

 

[Mark44]

Homework Statement:: Give an example of irrational numbers a
and b such that the indicated expression is (a) rational and (b) irrational.
Relevant Equations:: None.

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

1. a +b

2. a•b

3. a/b

4. a - b

What exactly is this question asking for? Can someone rephrase the statement above?

Thanks

The four parts of this question are pretty clearly stated. Perhaps you don't know what the terms "rational" and "irrational" in the context of numbers. Can you give a couple examples of rational numbers? A couple more of irrational numbers?

Does your textbook provide definitions and examples of these?

 

[DaveE]

OK, so there are 8 questions here. I'll only do the first two, the other 6 are pretty obvious, it's only the arithmetic operation that changes:

1a) Find two numbers a and b, each irrational, such that a+b is a rational number.
1b) Find two numbers a and b, each irrational, such that a+b is an irrational number.
---

I don't think it can be expressed in a simpler form. You will need to know what rational and irrational numbers are.

 

[nycmathguy]

OK, so there are 8 questions here. I'll only do the first two, the other 6 are pretty obvious, it's only the arithmetic operation that changes:

1a) Find two numbers a and b, each irrational, such that a+b is a rational number.
1b) Find two numbers a and b, each irrational, such that a+b is an irrational number.
---

I don't think it can be expressed in a simpler form. You will need to know what rational and irrational numbers are.

The sum, difference, product, and quotient of
an irrational number and a nonzero rational are all irrational. I will be back with my work when I find two irrational numbers such that when added produce a sum that is rational and irrational.

 

[nycmathguy]

OK, so there are 8 questions here. I'll only do the first two, the other 6 are pretty obvious, it's only the arithmetic operation that changes:

1a) Find two numbers a and b, each irrational, such that a+b is a rational number.
1b) Find two numbers a and b, each irrational, such that a+b is an irrational number.
---

I don't think it can be expressed in a simpler form. You will need to know what rational and irrational numbers are.

For a + b:

Irrational + Irrational = Rational

Let a = (1 - pi).

Let b = pi

a + b = (1 - pi) + pi

a + b = 1, which is rational.

Irrational + Irrational = Irrational

Let a = sqrt{2} = b

a + b = sqrt{2} + sqrt{2}

a + b = 2•sqrt{2}, which is Irrational.

Is this correct so far? If so, I will then move on to do the same for a•b, a/b l, and finally to a - b. I'll wait for your reply.

 

[DaveE]

looks good to me!

 

[nycmathguy]

looks good to me!

Ok. I will work on the rest and come back when time allows.

 

[nycmathguy]

The four parts of this question are pretty clearly stated. Perhaps you don't know what the terms "rational" and "irrational" in the context of numbers. Can you give a couple examples of rational numbers? A couple more of irrational numbers?

Does your textbook provide definitions and examples of these?

Let me do the a•b case.

Irrational • Irrational = Rational

Let a = sqrt{6} = b

a • b = sqrt{6} • sqrt{6}

a • b = sqrt{36}

a • b = 6, which is rational.

Irrational • Irrational = Irrational

Let a = sqrt{2}

Let b = sqrt{3}

a • b = sqrt{2} • sqrt{3}

a • b = sqrt{6}

You say?

 

[Mark44]

Let me do the a•b case.

Irrational • Irrational = Rational

Let a = sqrt{6} = b

a • b = sqrt{6} • sqrt{6}

a • b = sqrt{36}

a • b = 6, which is rational.

Irrational • Irrational = Irrational

Let a = sqrt{2}

Let b = sqrt{3}

a • b = sqrt{2} • sqrt{3}

a • b = sqrt{6}

You say?

Yes to both.

 

[nycmathguy]

The four parts of this question are pretty clearly stated. Perhaps you don't know what the terms "rational" and "irrational" in the context of numbers. Can you give a couple examples of rational numbers? A couple more of irrational numbers?

Does your textbook provide definitions and examples of these?

Let me do the a/b case.

Irrational ÷ Irrational = Rational

Let a = 2pi

Let b = pi

a/b = 2pi/pi

a/b = 2, which is rational.

Irrational • Irrational = Irrational

Let a = sqrt{2}

Let b = sqrt{3}

a/b = sqrt{2}/sqrt{3}

a/b = [sqrt{2}/sqrt{3}][sqrt{3}/sqrt{3}]

a/b = sqrt{6}/3, which is Irrational.

Yes?

 

[nycmathguy]

OK, so there are 8 questions here. I'll only do the first two, the other 6 are pretty obvious, it's only the arithmetic operation that changes:

1a) Find two numbers a and b, each irrational, such that a+b is a rational number.
1b) Find two numbers a and b, each irrational, such that a+b is an irrational number.
---

I don't think it can be expressed in a simpler form. You will need to know what rational and irrational numbers are.

Let me do the a - b case.

Irrational - Irrational = Rational

Let a = (2 + 2pi)

Let b = -2pi

a - b = (2 + 2pi) - 2pi

a - b = 2, which is rational.

Yes?

Irrational - Irrational = Irrational

Let a = pi

Let b = e

a - b = pi - e, which is Irrational.

You say?

 

[Mark44]

Irrational • Irrational = Irrational

Let a = sqrt{2}

Let b = sqrt{3}

a/b = sqrt{2}/sqrt{3}

a/b = [sqrt{2}/sqrt{3}][sqrt{3}/sqrt{3}]

a/b = sqrt{6}/3, which is Irrational.

You said your were doing the irrational * irrational case, but actually did the irrational / irrational case.

Let me do the a - b case.

Irrational - Irrational = Rational

Let a = (2 + 2pi)

Let b = -2pi

a - b = (2 + 2pi) - 2pi

a - b = 2, which is rational.

Yes?

Yes, fine.

Irrational - Irrational = Irrational

Let a = pi

Let b = e

a - b = pi - e, which is Irrational.

You say?

Right.

 

[PeroK]

Homework Statement:: Give an example of irrational numbers a
and b such that the indicated expression is (a) rational and (b) irrational.
Relevant Equations:: None.

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

1. a +b

2. a•b

3. a/b

4. a - b

What exactly is this question asking for? Can someone rephrase the statement above?

Thanks

At this level of mathematics $+$ and $-$ are essentially the same operation: subtracting a number is adding its additive inverse: $$a - b \equiv a + (-b)$$ Likewise, $$a/b \equiv a \cdot b^{-1}$$ Essentially, therefore, the same examples may be used for parts 1 & 4 and for parts 2 & 3.

 

[nycmathguy]

At this level of mathematics $+$ and $-$ are essentially the same operation: subtracting a number is adding its additive inverse: $$a - b \equiv a + (-b)$$ Likewise, $$a/b \equiv a \cdot b^{-1}$$ Essentially, therefore, the same examples may be used for parts 1 & 4 and for parts 2 & 3.

Good to know.