1What is the first step in a proof by contradiction?
Assume the statement is not true
2What statement are we proving with proof by contradiction?
“If x² is even, then x must be even”
3What happens if we assume the statement is false?
There must be an odd number for x for which x² is even
4How can an odd number x be expressed?
as x = 2k + 1, where k is an integer
5What is the expression for x² when x is odd?
x² = (2k + 1)² = 4k² + 4k + 1
6What does the expression 4k² + 4k represent?
it is even because it is 2 times an integer
7What conclusion do we reach if x² is even?
It contradicts the assumption that there is an odd number x for which x² is even
8What can be concluded about x if x² is even?
x must be even, hence the original statement is true
9What is the assumption made to prove that surds are irrational?
Assume that √2 can be written as a with a and b both non-zero integers
10What assumption is made about a and b in the proof involving √2?
a and b do not have any common factors
11What does the equation √2 = a imply?
√2b = a
12What do we get when we square both sides of √2 = a?
2b² = a²
13What can be concluded about a² in the proof regarding √2?
a² is an even number