Prove that if a is a natural number and the square root of a is
rational, then it is a square number (an integer n^2 for some
Stage: 5 Challenge Level:
To prove a theorem directly we start with something known to be
true and then proceed, making small logical steps which are clearly
correct, until we arrive at the desired result. So, because the
starting point was true and each small step clearly correct, we
know the result to be true.
Breaking down a mathematical argument into small steps requires
patience and clear thinking.
In the following interactivities we have written out three proofs,
broken them into small steps and then
shuffled up the steps. Can you rearrange them into the correct
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.