A number joined to a root
√3 is irrational. What about 4 + √3?
4 + √3 = p/q. Here p and q are whole numbers, and q is not 0. Step 1 of 4
- Suppose
4 + √3 = p/q. Here p and q are whole numbers, and q is not 0. - Move the 4 across. Then
√3 = p/q − 4. p/q − 4 = (p − 4q)/q. That is a fraction.- So √3 would be a fraction. But √3 is irrational. The guess was wrong.
4 + √3 is irrational too.
The rule
Suppose the number is a fraction. Get the root alone. The root would be a fraction.
That contradicts the root being irrational. So the number is irrational too.
The steps use one fact. Add, subtract, multiply or divide fractions, and you get a fraction. The one exception is dividing by 0.
| Number | Suppose | Root alone | Why that is a fraction |
|---|---|---|---|
| 4 + √3 | = p/q | √3 = p/q − 4 | (p − 4q) over q |
| 6√5 | = p/q | √5 = p/(6q) | p over 6q, and 6q is not 0 |
| 1/√3 | = p/q | √3 = q/p | q over p, and p is not 0 |
The proof needs the root alone to be irrational. √2, √3 and √5 are.
Watch out!
Check the root first. √9 is 3. So 2 + √9 = 5, a whole number.
The rule can fail in a second way. A zero multiplier breaks it. 0 × √3 = 0, which is rational.
Now try it. Tap Irrational or Rational.
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