Now that we've met the naturals, integers and rationals, only the irrational numbers are left to complete the collection of real numbers. These are the numbers that the Pythagoreans refused to believe could exist. Each of the new numbers demand a little more imagination than the last, to fully understand and appreciate the irrationals we have to contend with some tricky reasoning, but the rewards are worth it. In school maths lessons, the word 'surd' is often used in place of irrational, referring to the square root of any number. To be completely accurate, the word surd refers to a specific kind of irrational number, roots which are also irrational. The square root of 4 for example is not really a surd because it is equal to the rational number 2, but the square root of 5 is a surd, as is the cube root of 5. We can consider lots of different roots to find surds and, delightfully, there are lots of other kinds of irrational numbers too. The actual word 'surd' has roots (get it? - hilarious) in Latin, Greek and Arabic, it can be translated as 'deaf, mute, speechless or irrational.' These impossible roots earned their name by so frustrating the ancient world that they were declared 'deaf to reason' which conjures a pleasing picture of the Pythagoreans desperately trying to force a rational argument on these numbers that they just refuse to hear.
Deaf to Reasoning
This video introduces the concept of proof by contradiction and uses it to prove the irrationality of root 2. This form of proof can be used to show that lots of different roots are irrational.
Root 2 and all of the other irrationals are added to the rational numbers to complete the set of real numbers, R.
As well as causing trouble with fractions and decimals, the irrational numbers lead us to some mind-bending conclusions when it comes to the numberline which now must include all of the reals.
Watch the below video to find out a bit more about where and how the irrationals lie on the numberline, as well as what sorts of irrationals are out there in the complete set of reals.
When is a surd not a surd?
While there are many reasons why we might meet root 4 in the wild, were we to run home and announce we'd encountered a surd, the news that it was root 4 would undoubtedly disappoint and we may well be accused of 'crying wolf' when we should instead have cried '2'. The fact that root 4 is equal to the rational number 2 means that it is not a true surd. The same goes for the square root of any square, the cube root of any cube and so forth, as all have a rational value and so do not warrant the name. Furthermore, in mathematics we tend to call things by their most useful name for the purpose in hand, sometimes we think of three quarters as three quarters, sometimes 0.75, sometimes 75%, sometimes six eighths, etc. depending on what we're doing with it. Root 3 is a useful name for root 3 because it tells us that it will square to 3 and since it is irrational the name root 3 is certainly more useful than any rounded decimal or fraction approximation as all would be inaccurate. Conversely, we already know that 2 will square to 4 and there are just more useful things about 2, so it being the square root of 4 doesn't merit the headline.
These rationals masquerading as surds actually make our lives much easier because, if we can spot them, we are able to simplify which makes manipulating these irrationals far more straight forward. The mathematics of this relies on the multiplicative nature of surds, so to see how they behave it is not surprising that we look again to the law of commutativity.
Surds and Squares and Almost Theres
Now that we know how to manipulate surds, we should expect that some interesting things happen when we create different situations involving their multiplication with one another and with rationals. As with all new things in mathematics, we should begin by playing with it, so that we can see what happens and what things are at work. Perhaps we will stumble upon something useful or interesting, perhaps we will just have tremendous fun.
Just before we begin we should save ourselves from an error that is the downfall of so many who come to play with these irrationals.
We have recently proved these cheerful facts:
So one might almost be forgiven for assuming that:
Almost forgiven, not actually forgiven though because one must prove it to use it and those two statements are provably false. To convince yourself, write out, expand and simplify the following pairs of brackets:
You should find that:
So they almost make it there, but not almost enough. In any case where m and n are not both zero (making the exercise trivial) we can conclude that:
Now then, that's one mistake you'll never make.
Dearly Irrational, we are gathered here today...
A more profound consequence of this phenomenon is that it allows us to make the irrational rational, simply by choosing the appropriate multiplier. Whenever we are given (a+b) we choose (a-b) and vice versa. These partners are called 'conjugates' (specifically binomial conjugates but conjugate is enough for now) a word derived from Latin, translated as 'yoked together'. A yoke is a piece of wood used to bind pairs of oxen or other animals to enable them to pull one load together. We get lots of words from this same Latin root which all concern the coming together of a couple, like conjugal which describes matters relating to marriage. These conjugates are so named then to depict that they work together in their pairs, coming together in a marriage of multiplication to enjoy a lifetime of rationality. Aw.
This is especially useful when we find ourselves with an irrational in the denominator of a fraction, since dividing by an irrational number does not feel like a particularly comfortable thing to do, fortunately this gives us a way out. By multiplying by one, but a sneaky one expressed in terms of the conjugate, we can multiply away the irrational terms in the denominator, making it rational. Hence the name 'rationalising the denominator'.
In the final four questions of the above exercise, you should notice that the irrational terms in the expansion sum to zero and so effectively disappear. The very last question shows us the general form and so serves as a proof of this occurrence. You should recognise this as the 'difference of two squares' encountered when expanding double brackets elsewhere.