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sologuitar

Member

Registered: 2022-09-19

Posts: 467

Sum of Rational & Irrational Numbers

College Algebra
Section R.1

Why is the sum of a rational and irrational number irrational?

Bob

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Registered: 2010-06-20

Posts: 10,198

Re: Sum of Rational & Irrational Numbers

Prove this by assuming the converse.

Suppose a rational + an irrational  is rational.

Re-arrange so that rational - rational = irrational.  It is easy to prove that r - r is also rational.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Sum of Rational & Irrational Numbers

Can you show me what you mean?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,198

Re: Sum of Rational & Irrational Numbers

Let a, b, c and d be integers.

ad - bc is an integer.  bc is an integer. => the result of subtracting one rational from another is another rational.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Sum of Rational & Irrational Numbers

You subtracted one fraction from another to show that the result will always be
rational.

Let me see.

(1/2) - (1/4) = 1/4

We know that 1/4 is a rational number.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,198

Re: Sum of Rational & Irrational Numbers

If you would like a bit of extra practice here's some proofs you could try.

Show that the sum (and product and division) of two rationals is also rational.

Show similarly that two irrationals, when combined using + - x or ÷  is also irrational.

We say the rationals are closed under these four operations and the irrationals are similarly closed.

Are the rationals closed when we apply the square root function?

(Harder) Show that any rational is either a terminating decimal (eg 1/4 = 0.25) or a recurring decimal (eg. 1/3 = 0.33333....)

Hence deduce that any irrational must have an infinite, non recurring decimal form.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Sum of Rational & Irrational Numbers

Thanks for the practice problems. I will explore each in greater detail on my days off.