A rational number can be written as a/b, where a and b are integers and b is not zero. By cancelling common factors, we may choose a and b to be coprime. That lowest-terms condition is essential in contradiction proofs.
Practise the algebraic chain: sqrt(p) = a/b implies b sqrt(p) = a, and squaring gives p b^2 = a^2. Every transformation must apply equally to both sides.
Recall that prime factorisation is unique apart from order. This lets us reason about which primes must divide an integer when they divide its square.
A calculator can display many non-repeating digits of sqrt(2), but a finite display cannot prove what happens forever. A proof must show logically that no fraction in lowest terms can equal the number.
A real number is rational if it can be expressed as a/b for integers a and b with b non-zero. A real number that cannot be expressed in this form is irrational. For proof, choose the fraction in lowest terms so gcd(a,b) = 1.
If p is prime and p divides a^2, then p divides a. In the prime factorisation of a^2, every prime occurring in a appears twice as often. Therefore a prime found in a^2 must already occur in a. The condition that p is prime is essential.
A contradiction proof follows a precise chain: assume the opposite of the required conclusion; express that assumption exactly; derive a consequence that conflicts with a stated condition or known fact; identify the contradiction; reject the assumption; state the required conclusion.
For an expression such as r + sqrt(p), assume the complete expression is rational. Rearrange to isolate sqrt(p). The other side remains rational because rational numbers are closed under the required operations. This contradicts the known irrationality of sqrt(p).
If 5 divides a^2, explain why 5 divides a.
A contradiction proof eventually shows that a and b share a prime factor. This contradicts something only if a/b was deliberately chosen in lowest terms. Therefore the proof must state gcd(a,b) = 1 before deriving the common factor.
Prove that sqrt(5) is irrational using contradiction.
Prove that 6 + sqrt(2) is irrational using the known irrationality of sqrt(2).
Complete the opening correctly: Assume sqrt(7) is rational. Then sqrt(7) = a/b, where a and b are ______, b is ______, and gcd(a,b) = ______. Explain why each condition is needed.
To prove an expression irrational by contradiction, choose between: assume it is irrational; assume it is rational; or assume it is an integer. Explain why only the logical opposite of the target conclusion begins the proof.
Decide whether each inference is justified: (a) 3 divides a^2, therefore 3 divides a; (b) 6 divides a^2, therefore 6 divides a. Identify the theorem condition that distinguishes them.
A proof has shown that both a and b are divisible by 3. Complete the final three statements: what earlier condition is contradicted, what assumption must be rejected, and what conclusion follows?
Use the full coprime contradiction structure to establish that sqrt(p) is irrational. For a rational combination of an already known irrational number, isolate the known irrational instead. Choosing the shorter valid route improves clarity without omitting justification.
A compact board answer must preserve the entire logic: contrary rational assumption in lowest terms; divisibility of numerator; divisibility of denominator; contradiction with coprimality; final conclusion. Do not replace the proof with a decimal approximation.
Assume the complete expression is rational, isolate the known irrational term, explain why the other side is rational, name the contradiction, and conclude. Each logical link should be written explicitly.
Assume the logical opposite. Use a fraction in lowest terms. Check that the divisor theorem is applied only to a prime. State exactly which condition is contradicted. Reject the assumption and write the required conclusion. For composite expressions, isolate a known irrational whenever possible.
(1) Explain why the prime condition matters in the divisor theorem. (2) Prove that sqrt(7) is irrational. (3) Prove that 4 + 3sqrt(7) is irrational, using the shortest valid route.