Advanced Root Extraction, I
This is a translation of an article that appeared in the September 2005 issue of Pythagoras.
What is ∛2, really? What is its definition? How do you know it exists?
If someone asks you what √2 is, what do you say? Not `the square root of 2', because that is saying the same thing but then in words. The correct answer is: it is an abbreviation of `that positive number whose square is equal to 2'. The next question might be about the value of √2. And then you will have to explain that √2 can not be expressed as a fraction and that √2 is really not more than an abbreviation. Then you get a tricky question: how do you know that it exists? Fortunately you can use a bit of geometry to point out the position of √2 on the number line.
Just make square whose sides have length 1 its diagonals then have length √2.
What is ∛2?
The question about ∛2 has a clear answer: `the positive number whose third power is equal to 2'. Does that number exist as well? Can we point out where it is on the number line? That depends of course on what you mean by `pointing out', but it is not as easily done as for √2. Not only is ∛2, like √2, not expressible as a fraction but there is even no geometric construction with ruler and compasses that will give a line segment of length ∛2.
We can of course try to point out where ∛2 is approximately. If you set a calculator to work you will soon discover that 1.23<2<1.33, so that ∛2 lies in the interval (1.2,1.3). If you keep going you will find 1.253<2<1.263, … 1.25598921043<2<1.25598921053, and so ∛2 lies in the interval (1.2559892104,1.2559892105). If we keep going we shall make ever smaller intervals that should contain ∛2. And in this way we seem to find ever more digits in the decimal expansion of ∛2, but at no point do we actually have ∛2 itself and it is not certain whether ∛2 actually exists. After all it could be that in between all those approximations there is a gap in the number line and that we have been calculating with something that is not there.
Fortunately that is not the case. There are no gaps (or holes) in the number line and that is a property that makes basically all of Mathematical Analysis possible. Thanks to that property we can show that ∛2 exists. The beauty of the property and the proof is that you can show in the same way that all nth roots of 2 exist. That makes the property so very important: it can be used to prove the existence of very many numbers all at once.
A fundamental property
The mathematically precise formulation of "the number line has no holes" is "the number line is complete".
To formulate what completeness means we need the concept of upper bound.
Suppose A is a set of numbers; a number x is an upper bound of A if a≤x holds for all a in A. For example 1000 is an upper bound of the interval [0,1) and 2 is an upper bound of {x:x2<2}. Of course 1000 is a very crude upper bound of [0,1), there are better ones, 10 is also an upper bound, as is 1. We cannot go below 1 of course: 1 is the least upper bound of [0,1).
When you draw (or write down) more examples of sets of numbers you will see that whenever there is an upper bound there is always also a least upper bound. This property holds for all sets (even if their description is not so simple).
Theorem Every nonempty set of numbers that has an upper bound also has a least upper bound.
A proof of this fundamental property is not that easy; you can give one if you know exactly how the real numbers are made. We save that for a later post.
Now we will show how you can use completeness to prove the existence of ∛2.
The existence of ∛2
Consider the set A={x:x≥0 and x3<2}. That set is not empty because 0 and 1 belong to A; the set has an upper bound, 2 for example. Now we can conclude that there is a number α that is the least upper bound of A, so x≤α for all x∈A (as α is an upper bound), and if $β<&alpha then there is an x∈A such that x>β (as α is the smallest upper bound). We shall prove α3=2 and hence that we established the existence of ∛2.
We need a formula: for all numbers x and y we have
y3-x3=(y-x)(y2+xy+x2)
You can verify this by expanding the right-hand side. Using this formula we can find everything we need.
To begin: if 0≤x<y then x3<y3. This follows because y2+xy+x2 is positive. From this we deduce that once we have shown that α3=2 we know that there is no other number with that property.
Next: if 0<x<y<2 then
y3-x3<(y-x)(4+4+4)=12(y-x)
In particular: if x∈A then
α3<x3+12(α-x)<2+12(α-x)
Because we can make α-x as small as we please it follows that α3≤2.
Third: if α<y<2 then
alpha3>y3-12(y-α)≥2-12(y-α)
(because y is not in A we have y3≥2). Again we can make y-α as small as we please and deduce that α3≥2.
Conclusion: α3=2.
Remarks
In the same way we can show that every nth root of 2 exists, for every n, as the least upper bound of the set
An={x:x≥0 and xn<2}
There is nothing special about 2 of course; we can use the above arguments to prove the existence of the nth root of a for every positive number a and every natural number n.
Next time we will discuss an efficient way of making approximations of those nth roots.
