How can we use irrational numbers in math if we don’t know the full number?
In: 232
You don’t need to. The resulting number may be infinitely long and chaotic, but nobody needs to see all the digits. Whatever you’re measuring, the tool has some kind of limit and at some point you might as well stop in your calculations. For most people, the diameter of a circle (which involves the irrational number pi) to 0.01 microns is probably more than enough detail.
Also, it’s very common to leave the number in symbolic form until such time as you actually need a decimal number. The square root of 2 is irrational, but while you’re doing your math, you just leave it as `sqrt(2)` as if this were some unknown variable like `x`. When you’re done and you’ve made the final formula as simple as you can, you can actually substitute in `1.41421356237309504880`, or stop at however many decimal places make sense, for your final numeric answer.
3.14159265358979323846264338327950288420
These are all the digits of Pi necessary to calculate the circumference of the visible universe to a precision of a hydrogen atoms diameter. As you can see, it is far from infinite digits.
And to get a bit more philosophical, while infinities exist in math, they do not seem to exist physically in our universe, so infinite precision is not only not necessary, it’s probably not possible. So worrying about the “full” number is pointless. And in the math itself you just dismiss the issue by giving the number a symbol and calling it a day.
By this logic, we don’t know the full number of 1/7 either. It has an infinite decimal expansion, it goes on forever. But we can calculate it to whatever precision we want because we know the pattern.
This is how we use irrational numbers. The patterns are a bit more complicated, but we still have formulas or algorithms to approximate the value of these numbers as closely as we want to get it.
We still know enough about the number to do useful things.
Eg, the square root of 2 is irrational. We know it squares to 2. From that, we can figure out that the square root of 2 is between 1.41421 and 1.41422, so if I want to cut a piece of wood to a length of sqrt(2) metres, I can do so to way better than millimeter precision. If I need better precision still, I can always work out more decimal places of sqrt(2).
We know the numbers, they simply don’t match with representing them in other numerical forms. Pi is a classic example, it is the circumference of a circle divided by its diameter. That’s pi. But that ratio doesn’t lend itself to portion of 10 base.
These answers are all entirely missing the point.
To illustrate, try the number
0.1234567891011121314…
Its irrational as you can get (transcendental and normal), and yet I don’t think you’d argue that we don’t “fully know the number”.
In general, for (computable) irrational numbers, we *do* know the full number. The fact that we cannot faithfully represent an irrational number in decimal notation doesn’t mean we “don’t know the full number”, it means decimal notation is an awfully inconvenient way to represent irrational numbers.
Zooming out further, it’s unclear to me why the symbols √2 or π are any less “exact” than the symbols 3, or 7/4. Each of those symbols specify a unique real number. Again, it’s true that in decimal notation irrational numbers cannot be captured by a finite string of digits, but base notation is a fairly artificial way of representing a number, so this doesn’t say much about the fundamental nature of irrational numbers.
We do know the full number, we just don’t (can’t) know the full decimal that represents that number.
Pi is pi. It’s the ratio of a circle’s circumference to its diameter. 3.14159.. isn’t pi, that’s just an approximation that gets pretty close.
sin(pi) = 0, by definition. sin(3.14159) = 0.00000265.. (at least according to Google).
10 + pi is approximately 13.14159, sure. But ‘10 + pi’ describes the number more accurately and precisely. If we really need to, like we’re doing maths to design a machine or something, we can use the approximation at the end – it’s a lot simpler to tell a machinist to make it 13.14cm long than it is to tell them to make it (10 + pi) cm long. But at that point, it’s stopped being pure mathematics, and so approximations within a certain tolerance are more than good enough.
Because real life isn’t that precise.
This was something I had to get over when I went into Engineering back in the early 2000s. Significant digits. Those things you learned in high school actually mean something. So if you are measuring out some wood and you got a tape measure, the best you can measure is going to be 0.001 meters. Since means that any other number you use past 3 decimal points is worthless since you don’t know the 4th and on for the length. It looks like 1mm so you use 1mm. Not 1.0 which means you know its 1 mm down to the tenth. 1.00 means you know its 1mm down to the hundredth of a mm giving you more and more significant digits.
So if you got 0.001m (1mm), pi past 3.142 is useless because your primary measurement you are working with is only significant to 1mm. Any decimal past that should be truncated and rounded. So you did some math and you used 3.141592 for pi, you end up with a final value for 0.0031233 as your answer, its is only good to 0.003.
We can use √2 because we know that √2 × √2 = 2. Sometimes we don’t even need to calculate it out in decimal; because the √2’s end up cancelling out.
We also know that √2 is the length of the hypotenuse of a right triangle whose sides have length 1, because of the Pythagorean theorem: a² + b² = c², so if a = b = 1, then c = √2.
That means that if we can measure a right angle and a side of length 1, we can measure something of √2 length without ever having to have a √2 mark on our ruler.
We do know the full number!
For example, the square root of 2 is an irrational number, but “√2” is a complete and usable description of the number.
You can add it (√2 + √2 = 2√2),
multiply (2√2 × 3√2 = 12),
divide (√2/2),
raise to a power (√2^2 = 2), or anything else!
Try for yourself!
Oh, you must mean we can’t see the end of the decimal representation of it! 1.41421356… it just keeps going and doesn’t repeat! Not a problem, just write √2.
Sometimes, an estimate is good enough. Let’s say I want to make a table big enough so that an XL pizza box (16 inches on the side) could fit on the table without hanging over the edge. Such a table requires a diameter of 16√2 inches, or roughly 16*1.4 = 22.4, let’s call it 23 inches. So, if I cut a circle out of wood that’s 23 inches wide, an XL pizza box can spin around at the center, and not a corner would hang off. Here, √2 was perfectly usable even using a “rational approximation,” like 1.4 (7/5).
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