From exploration to demonstration

Finding √2 in ranges helps to visualize. The demonstration of irrationality requires the mathematical argument presented in this class.

Explore the controls and explain what changed in your own words.

Why can’t √2 be a fraction?

Suppose, to obtain a contradiction, that √2 = a/b, with a and b positive integers and the irreducible fraction. Then a² = 2b², so a² is even. If a were odd, its square would be odd; therefore a is even. Write a = 2k. Substituting, 4k² = 2b², or b² = 2k², which also makes b even.

Thus, a and b would be divisible by 2, contradicting the hypothesis of an irreducible fraction. Therefore, √2 is irrational. The decimal approximation was not used as proof.

Operations: investigate before generalizing

Operation Result Classification
√2 + (−√2) 0 Rational
√2 × √2 2 Rational
1 + √2 1 + √2 Irrational
2√2 2√2 Irrational

If r is rational and x is irrational, r + x is irrational: if the sum were rational, subtracting r would make x rational, a contradiction. If r ≠ 0, the same reasoning, using division, shows that rx is irrational. The condition r ≠ 0 is required: 0 × x = 0.

Logarithms challenge

Is log₂8 irrational? And log₂3?

Check the justification

log₂8 = 3 is rational. If log₂3 = p/q were rational, with p and q positive integers, we would have 2ᵖ = 3ᑫ. This is impossible: the left side is even and the right is odd. Therefore, log₂3 is irrational.

Continue to the logarithms.