Worked examples
The question
Question: prove that √2 is irrational.
1 Assume the opposite
Rational means: it can be written as a fraction of whole numbers.
Irrational means it cannot.
rational = a fraction ab of whole numbers
Suppose the opposite: √2 is rational.
Then √2 = a/b.
√2 = ab
We cancel the fraction as far as it goes: a and b have no common factor.
in lowest terms:
a and b have no common factor
Careful!
Without lowest terms there is nothing to break.
Remember it: a and b have no common factor.
Careful: lowest terms is the key
2 Work it out
Square both sides: √2 squared is 2, and a/b squared is a squared over b squared.
2 = a2b2
Times b squared on both sides: a² = 2b².
a2 = 2b2
2 times a whole number: so a squared is even.
And when a squared is even, a is even.
We proved that by contrapositive.
a² is even
n² even → n even (u74)
So a is even too: a = 2c, with c a whole number.
a = 2c
Put it into a squared = 2 b squared.
(2c)2 = 2b2
2c squared is 4c squared.
4c2 = 2b2
Divide both sides by 2: b² = 2c².
b2 = 2c2
The same as before: b squared is even, so b is even.
b² is even, so b is even
3 Find the contradiction
But now a and b are both even.
2 is a common factor.
That breaks lowest terms!
a and b are both even: 2 goes into both ✗
The opposite led to nonsense, so it is false.
√2 is irrational.
So √2 is not rational: √2 is irrational. ∎
4 Check
A calculator: √2 is about 1.41421356, and the digits go on with no repeating pattern.
√2 ≈ 1.41421356...
Even a good fraction like 17/12 squares to about 2.007, close to 2 but not 2.
Right!
(1712)² = 289144 ≈ 2.007 ≠ 2
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.
Answer So √2 is not rational: √2 is irrational. ∎
▶ Watch it solved
The question
Question: prove that √7 is irrational.
1 Assume the opposite
Rational means: it can be written as a fraction of whole numbers.
Irrational means it cannot.
rational = a fraction ab of whole numbers
Suppose the opposite: √7 is rational.
Then √7 = a/b.
√7 = ab
We cancel the fraction as far as it goes: a and b have no common factor.
in lowest terms:
a and b have no common factor
Careful!
Without lowest terms there is nothing to break.
Remember it: a and b have no common factor.
Careful: lowest terms is the key
2 Work it out
Square both sides: √7 squared is 7, and a/b squared is a squared over b squared.
7 = a2b2
Times b squared on both sides: a² = 7b².
a2 = 7b2
7 times a whole number: so a squared is a multiple of 7.
7 is a prime number.
When a prime goes into a times a, it goes into a.
a² is a multiple of 7
prime 7 into a × a → into a
So a is a multiple of 7 too: a = 7c, with c a whole number.
a = 7c
Put it into a squared = 7 b squared.
(7c)2 = 7b2
7c squared is 49c squared.
49c2 = 7b2
Divide both sides by 7: b² = 7c².
b2 = 7c2
The same as before: b squared is a multiple of 7, so b is a multiple of 7.
b² is a multiple of 7, so b is a multiple of 7
3 Find the contradiction
But now a and b are both a multiple of 7.
7 is a common factor.
That breaks lowest terms!
a and b are both a multiple of 7: 7 goes into both ✗
The opposite led to nonsense, so it is false.
√7 is irrational.
So √7 is not rational: √7 is irrational. ∎
4 Check
A calculator: √7 is about 2.64575131, and the digits go on with no repeating pattern.
√7 ≈ 2.64575131...
Even a good fraction like 29/11 squares to about 6.950, close to 7 but not 7.
Right!
(2911)² = 841121 ≈ 6.950 ≠ 7
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.
Answer So √7 is not rational: √7 is irrational. ∎
▶ Watch it solved
The question
Question: prove that √13 is irrational.
1 Assume the opposite
Rational means: it can be written as a fraction of whole numbers.
Irrational means it cannot.
rational = a fraction ab of whole numbers
Suppose the opposite: √13 is rational.
Then √13 = a/b.
√13 = ab
We cancel the fraction as far as it goes: a and b have no common factor.
in lowest terms:
a and b have no common factor
Careful!
Without lowest terms there is nothing to break.
Remember it: a and b have no common factor.
Careful: lowest terms is the key
2 Work it out
Square both sides: √13 squared is 13, and a/b squared is a squared over b squared.
13 = a2b2
Times b squared on both sides: a² = 13b².
a2 = 13b2
13 times a whole number: so a squared is a multiple of 13.
13 is a prime number.
When a prime goes into a times a, it goes into a.
a² is a multiple of 13
prime 13 into a × a → into a
So a is a multiple of 13 too: a = 13c, with c a whole number.
a = 13c
Put it into a squared = 13 b squared.
(13c)2 = 13b2
13c squared is 169c squared.
169c2 = 13b2
Divide both sides by 13: b² = 13c².
b2 = 13c2
The same as before: b squared is a multiple of 13, so b is a multiple of 13.
b² is a multiple of 13, so b is a multiple of 13
3 Find the contradiction
But now a and b are both a multiple of 13.
13 is a common factor.
That breaks lowest terms!
a and b are both a multiple of 13: 13 goes into both ✗
The opposite led to nonsense, so it is false.
√13 is irrational.
So √13 is not rational: √13 is irrational. ∎
4 Check
A calculator: √13 is about 3.60555128, and the digits go on with no repeating pattern.
√13 ≈ 3.60555128...
Even a good fraction like 18/5 squares to about 12.960, close to 13 but not 13.
Right!
(185)² = 32425 ≈ 12.960 ≠ 13
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.
Answer So √13 is not rational: √13 is irrational. ∎
▶ Watch it solved
