Rational and irrational mix

The idea

The sum or difference of a rational number and an irrational number is irrational.

5 minus root 3

We know √3 is irrational. Take away from the rational number 5:

Is 5 − √3 rational or irrational?

• Rational
• Irrational
• Not sure

Test it by contradiction

Assume 5 − √3 is rational, then climb. Watch the tags: rational or irrational.

Rational ± irrational

The sum or difference of a rational and an irrational number is irrational.

Why? Rational numbers stay rational when you add or subtract them. If 5 − √3 were rational, then 5 − (5 − √3) = √3 would be rational too. That is false.

Be careful: the rule is for a rational with an irrational. Two irrationals can add up to a rational: √2 + (−√2) = 0.

Notes

The sum or difference of a rational number and an irrational number is irrational. (If 5 − √3 were rational, √3 would be rational.)

Check yourself

Which of these is irrational?

Why does the proof work?

• √2 − √2. √2 − √2 = 0, which is rational. Both parts are irrational, so the rule does not apply.
• (2 + √3) + (2 − √3). This equals 4, which is rational.
• 3 + √5 — correct. Yes! A rational number plus an irrational number is irrational.
• 5 − 3. 5 − 3 = 2, rational.
• If a rational is added to or subtracted from a rational, the result is rational — correct. Yes! So an assumed-rational 2 + √7 would make √7 rational.
• Because √7 is a whole number. √7 is not a whole number. It is irrational.
• Because 2 + √7 is a decimal. Being a decimal is not the reason. The reason is that rationals stay rational under + and −.