# I How did the Greeks deal with sqrt(2)?

#### musicgold

I read the story of Hippasus's proof which showed $\sqrt {2}$ is neither a natural number nor a ratio of natural numbers and how he was drowned by the Pythagoreans for his impiety.

My question is as the Greeks did not know about decimals (until a few centuries later), how did they deal with irrational numbers when they presented themselves in their calculations and measurements.

How do you deal with a quantity that can't be represented using existing methods? Did they just drop those calculations or approximate the number?

Thanks

#### brainpushups

I haven't read the text yet, but this is the subject of Book X of Euclid's Elements, the content of which is most often attributed to Theaetetus. It is the longest book in the Elements and, according to Victor Katz (author of A History of Mathematics) it is the most well-organized. You can read a translation here which includes a guide to many propositions.

I'm currently at the end of book VIII so I'll hopefully be there soon...

#### QuantumQuest

Gold Member
I read the story of Hippasus's proof which showed $\sqrt {2}$ is neither a natural number nor a ratio of natural numbers and how he was drowned by the Pythagoreans for his impiety.

My question is as the Greeks did not know about decimals (until a few centuries later), how did they deal with irrational numbers when they presented themselves in their calculations and measurements.

How do you deal with a quantity that can't be represented using existing methods? Did they just drop those calculations or approximate the number?

Thanks
The story about Hippasus is an interesting account but its historical accuracy, as far as I know, is not certain. In any case, it turns out that Greeks knew about irrational numbers - see also Zeno's paradox. Pythagoreans just didn't like the idea because it was against their philosophy that natural numbers and their ratios describe the physical laws. Now, again as far as I know, Pythagoreans indeed try to approximate the value of $\sqrt{2}$ with high precision but unsuccessfully - for instance see the relevant table that depicts their method at ancient.eu.

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#### brainpushups

I read the story of Hippasus's proof which showed $\sqrt {2}$ is neither a natural number nor a ratio of natural numbers and how he was drowned by the Pythagoreans for his impiety.

My question is as the Greeks did not know about decimals (until a few centuries later), how did they deal with irrational numbers when they presented themselves in their calculations and measurements.

How do you deal with a quantity that can't be represented using existing methods? Did they just drop those calculations or approximate the number?

Thanks
Note that the OP's question is about 'dealing' with irrational numbers. There is no question that the ancient Greeks knew about irrational/incommensurable numbers.

So, if by 'deal' we mean 'approximate them for practical application' (for no one would need an exact irrational number for practical constructions or calculations) then there was no challenge for them in doing this. One method they were probably aware of for doing so was the Babylonian Method. Another method that could be used is the Euclidean Algorithm (Propositions VII.1 and VII.2 of the Elements). This algorithm finds the greatest common divisor between two numbers. When applied to incommensurable magnitudes one finds that the method does not terminate and this would provide a way to approximate the value of an irrational number. However, my understanding is that the ancient Greeks were not concerned much with calculation. They were more interested in determining rational, universal truth.

So, if by 'deal' we mean 'conceptualize the existence of' then that is what Book X of the Elements is probably a good reference for this. But, even in the earlier books of the Elements Euclid has to 'deal' with irrationals, and this is all done geometrically. For example, take the construction of the Mean Proportional between any two straight lines. Such a construction will often result in incommensurable lengths. Euclid gives a proof of this construction at least two times in the Elements (end of book II is the first appearance, with a more elegant one given later that the link above summarizes).

#### Michael Hardy

You seem to be working under a strangely widespread and persistent confusion about what irrational numbers are. There is nothing in the definition of irrational numbers that says anything at all about decimals or decimal digits.

DEFINITION: An irrational number is a number that is not a quotient of two integers.

For example, $12/7$ is rational since $12$ and $7$ are integers.

The most well known ancient Greek proof of the irrationality of $\sqrt2$ is this: If $\sqrt2 = \dfrac a b$ for some integers $a,b$, where the fraction $a/b$ is in lowest terms, then at least one of $a,b$ is odd. So $\left(\dfrac a b \right)^2 = 2$; hence $a^2 = 2b^2$; hence $a^2$ is even; hence, being a square, $a^2$ is a multiple of $4$; hence $2b^2$ is a multiple of $4$; hence $b^2$ is a multiple of $2$; hence $b$ is even. But this contradicts the fact that mentioned earlier, that at least one of them is odd. Hence the assumption that $\sqrt2$ is rational leads to a contradiction.

But the Greeks didn't consider non-integer lengths to be numbers; they didn't say that this number is irrational; rather than said the the lengths of the side and the diagonal of a square have no measure in common, i.e. no line segment would go into both of them some whole number of times.

"How did the Greeks deal with sqrt(2)?"

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