Question

Q5. Create an expression for each type of scenario below.

Rational times Rational

Rational plus Rational

Rational minus Rational

Rational divided by Rational

Rational times Irrational

Rational plus Irrational

Rational minus Irrational

Rational divided by Irrational

Irrational times Irrational

Irrational divided by Irrational

Irrational plus Irrational

Irrational minus Irrational

Answer 1

Here are one-line examples for each scenario, the result shown, and a short note about whether that result type is guaranteed or can vary.

1) Rational × Rational
Example: (2/3) × (9/4) = 3/2 — rational. (Always rational; denominator ≠ 0.)

2) Rational + Rational
Example: 1/2 + 3/4 = 5/4 — rational. (Always rational.)

3) Rational − Rational
Example: 5 − 2 = 3 — rational. (Always rational.)

4) Rational ÷ Rational
Example: (1/2) ÷ (3/4) = 2/3 — rational. (Always rational when divisor ≠ 0.)

5) Rational × Irrational
Example: 2 × √2 = 2√2 — irrational. (Usually irrational; exception: 0 × (irrational) = 0, which is rational.)

6) Rational + Irrational
Example: 3 + √2 — irrational. (Usually irrational; exception: specially chosen numbers can cancel and give rational.)

7) Rational − Irrational
Example: 5 − √3 — irrational. (Usually irrational; cancellation can produce rational in special cases.)

8) Rational ÷ Irrational
Example: 1 ÷ √2 = 1/√2 (equals √2/2) — irrational. (Usually irrational; exception: 0 ÷ (irrational) = 0.)

9) Irrational × Irrational
Example: √2 × √3 = √6 — irrational. (Can be either: e.g. √2·√2 = 2 is rational.)

10) Irrational ÷ Irrational
Example: √2 ÷ √3 = √(2/3) — irrational. (Can be either: e.g. √8 ÷ √2 = 2 is rational.)

11) Irrational + Irrational
Example: √2 + √3 — irrational. (Can be either: e.g. √2 + (2 − √2) = 2 is rational.)

12) Irrational − Irrational
Example: √3 − √2 — irrational. (Can be either: e.g. √2 − √2 = 0 is rational.)

If you want, I can give more examples that show the exceptional cases (where the result switches type).