Standard +0.8 This is a proof by contradiction requiring students to understand the definition of rational numbers and apply logical reasoning. While the proof structure is standard (assume the sum is rational, derive a contradiction), it requires formal mathematical thinking beyond routine calculation, making it moderately challenging for A-level students who may have limited experience with proof writing.
Question 9:
9 | Begins proof by contradiction.
This may be evidenced by: stating
assumption at the start “the sum is
rational”
Or
Sight of “contradiction” later as part
of argument. | 3.1a | M1 | Assume m is rational and n is
irrational and their sum is rational.
Then
a c
+n=
b d
Where a, b, c and d are all
integers.
c a
n= −
d b
bc−ad
=
bd
∴n is rational, which is a
contradiction.
So the original statement is false
and the sum of a rational and
irrational must be irrational.
Forms an equation of the form
rational + irrational = rational with
the rationals written algebraically
a c
+n=
b d
n must clearly be irrational and not
written as an algebraic fraction and
not a specific value. | 2.5 | M1
Manipulates their equation to show
that n is rational | 1.1b | A1
Explains or demonstrates why
there is a contradiction | 2.4 | E1
Completes rigorous argument to
prove the required result including
correct initial assumptions
Where a, b, c and d are all
integers. | 2.1 | R1
Total | 5
Q | Marking instructions | AO | Mark | Typical solution