We depend on zero for our math to work right, but this number was actually invented in fairly recent times. Why do we need zero? Was it inevitable?

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### Show Notes

The Origin of the Number Zero

Sumeria and Mesopotamia

Greek scholars and Ptolemy

Fibonacci coined term Zero

Definition of zero

Dividing by zero

Why dividing by zero is undefined

### Transcript

Transcription services provided by: GMR Transcription

Fraser: Astronomy Cast, Episode 466: The Origins of Zero (0). Welcome to Astronomy Cast, your weekly fact-based journey through the cosmos, where we help you understand not only what we know, but how we know what we know. My name is Fraser Cain. I’m the Publisher of Universe Today. With me is Dr. Pamela Gay, the Director of Technology and Citizen Science at the Astronomical Society of the Pacific, and the Director of CosmoQuest. Hey, Pamela, how are you doing?

Pamela: I’m doing well. How are you?

Fraser: Good. And, you are calling in from the country’s capitol. Your country’s capitol.

Pamela: I am. Yes, my country capitol. It’s far warmer than yours, I think, right now. Yeah, I’m near Dulles Airport at a meeting location for a collaboration meeting for everyone who receives the kind of funding that I get from NASA that is basically designed to allow us to better collaborate, keep up to date on what we’re all doing, and finding ways to prevent duplication of effort, and spread the effort better across all of us.

Fraser: Perfect. So, if you sound a little different, that’s because you’re recording on your portable audio device that’s sitting in front of you, from your hotel wifi. This is us being incredibly flexible to make this episode happen. So, I’m really glad you were able to sneak in some time to be able to make this happen. We depend on zero for our math to work right, but this number was actually invented in fairly recent times. Why do we need zero? Was it inevitable? Pamela, zero feels like one of those things that is just part of math, like it’s woven into the fabric of the cosmos. It is the middle point between the positive numbers and the negative numbers. But, it’s not?

Pamela: So, if you want to be literal about it, in the Mayan knot making wave recording accounting, it’s literally not a knot.

Fraser: It’s not a knot. What?

Pamela: So, one of the ways that the Mayans recorded numbers – and, you just stumbled into this, and gave me the best opening ever – is they would actually do accounting on ropes by tying specific kinds of knots. And, the way that they recorded a zero was to not have a knot there.

Fraser: Got it. So, it was a not knot. But, they didn’t know in ancient Mayan civilization that not and knot would be the same sounding word used hundreds of years hence in the English language. Man.

Pamela: It’s true. Unless there is a time-traveling punster. I’m really hoping for a time-traveling punster.

Fraser: Right. Okay. But this is just one example of a civilization coming up with this concept of zero. And, that’s not the only one. People have come up with it multiple times, separately, kind of like animals figuring out how to fly, right? Or, eyes, or things like that.

Pamela: And, they have had to come up with the concept of zero in multiple different ways. So, for instance, with the exception of the Egyptians – who were weird and had hieroglyphs for everything – most systems of numbering are base something. So, here in English language, using Latin alphabet, and the algebra that you and I both learned in school, we count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And, then it starts over – 11, 12 – and, these sets of numbers keep paralleling.

Fraser: I have the computer science background, so I just count 0, 1, 011, 0011 – you know. That’s all I know.

Pamela: I don’t believe you. If you were a true computer scientist, you’d work in hexadecimal.

Fraser: Wow. Thrown down. No, I still need a calculator to convert my hexadecimal into my binary code. Hold up. I actually learned binary code – It’s funny to totally rabbit-hole here – I learned binary code when I was a little kid. Someone taught me that you could count to 32 on your – just with one hand. So, you put out your hand, and you bring down one pinky, and that’s 1. Then you bring down your next finger, and that’s 2. And, then you bring down your pinky again, and that’s 3 – 4, 5, 6, 7, 8 – etc. So, you can actually count up in binary code on your hand. And, I ended up using that when I need to count. I still, sometimes, have to calculate with binary code. I use my hands.

Pamela: I think you just explained to me how valved instruments work in a way that finally made sense.

Fraser: If they had turned valved instruments into binary code, yeah, you could probably write it in binary code, and that would make a ton of sense. Anyway, I totally rabbit-holed this conversation. Let’s go back to talking about the development of zero.

Pamela: So, you need zero for three different positions. What I was starting to say is if you work in a base 10, a base 8, a base 20 – whatever base you work on, it turns out that when you repeat, you are periodically going to have a nothing. So, for instance, you have a situation where your nothing occurs. So, for instance, with 10 it goes 1,0. And, so what do you do when you have 104 of something? You have the 1’s digit. You have nothing in the tens’ digit, and then you have 4 in the ones’ – So, it’s 1 in the hundreds’; 0 in the tens’, 4 in the ones’. And, this combination – what do you do if you don’t have the concept of zero?

So, one of the original things that happened was they just basically put slashes there – angled slashes. And these two angled slashes denoted, “There’s nada in this place-holder.” But, that is a different concept from the concept of nothing/zero, or the transition from negative amounts to positive amounts – which you start to need when you get into algebra, and deficit accounting – which we should all be probably avoiding, and most of us aren’t, especially as we enter the giving season.

Fraser: So, why do we need zero? It was developed by the Egyptians. It was developed by the Mayans. It was developed – The Greeks had their own version of it.

Pamela: The Babylonians.

Fraser: Yeah. – The Chinese. The classic one that we use today was developed in the Middle East, right, by Islamic mathematicians. So, it was clearly this idea that needed to happen. What do we need it for?

Pamela: Go back to that place-holder reference. You’re doing accounting tables, and you’re keeping columns of numbers, and some poor slob is reading along later, trying to figure out, “How many chickens did you have?” And, the way you wrote 104 is 1 in the hundreds’ column; nothing at all – just a blank space – in that tens’ column – which is what they were initially doing. And, then a 4 in that ones’ space. Now, the problem is this poor slob who is trying to deal with this stuff later – he doesn’t know if you goofed and forgot to write something down, or you meant that you had nothing. So, that ambiguity is the kind of thing that breaks accountants, and mathematicians, and actuarials.

And, all of these people who are really worried about all of the numbers being right. And, for early math, it was accounting and actuarial work. You had to know how many people you were collecting taxes from. It was that where it really mattered. There is a difference between 194 chickens – if you forgot to write the 9 down –

Fraser: Right.

Pamela: – and 104 chickens. So, this is where that initial – just, slash/slash, stab it into your cuneiform – was actually quite useful.

Fraser: So, they would put a 1 for the 1 number, and whatever is the ancient Sumerian digit for 9 – they would do that because they were still using base 10 as well, right? But, they would put – like, to show that it’s a null digit – it’s not one of the nine digits – it’s a not digit, but don’t forget it.

Pamela: Right. And, so this is coming from the Babylonians, and it was just basically a, “We need something to shove in here to mean this place-holder has no values in it.”

Fraser: So, that’s one reason why we need a zero.

Pamela: That is one very good reason we need a zero. Now, the Egyptians who weren’t big on base anything, or having a written language that was based in an alphabet – they apparently liked to make is as hard as absolutely possible to be literate. And, in order to do this, the way they denoted zero was it had it’s own symbol, but so did every single other number. So, you had number, upon number, upon number. Basically, the way they referred to it is, “These are not positional.” It’s just a whole series of 1, 2, 3 – is a set of symbols. 13, 14, 15 – a set of symbols not related to one another.

You can see no resemblance between 4, 14, and 104, but these were architects. These were people who were very worried about geometry and measuring positions. So, they’re the ones who came with that concept of zero as that place where your are measuring your base from. That ground-level, that zero truth. And, so they had the concept of zero. They had the symbol for it. What they were missing was the positional concept. So, you have the Babylonians who are actually the ones who gave us that whole, crazy hours, minutes, seconds system. So, they were working in sexadecimal just to make life evil.

Fraser: Right. Like base 60 –

Pamela: Yeah.

Fraser: Which is crazy to have a circle be broken up into 360 degrees, and to have hours – Obviously, you know, have hours be broken up into some number, but have that broken up into minutes, and seconds, but also measuring. So, that’s all the Babylonian’s fault?

Pamela: Yes. Yes, it is.

Fraser: Mm-hmm.

Pamela: Now, the Egyptians, who did work in base 10, but didn’t have positional math – instead had these unique symbols to make it really hard to be literate, but made very pretty writing. Their very beautiful system had the, “Start here. This is where there is a nothing value.” And, what’s cool is the symbol that they used also meant, “Beautiful.” So, they considered nothing beautiful, which if you don’t pause and have a beat there, has a completely different meaning, but to them that concept of nothing was a beautiful thing, and that’s cool.

Fraser: Yeah. So, the Egyptians were using zero to try to help them with their pyramids, and whatever it is they were building. They Babylonians were very concerned about making sure they got all of their taxes in, so that they could fund their enormous armies, and conquer the nearby lands. Who else was working on zero?

Pamela: It was one of these things that cropped up just about everywhere in different ways. So, we had them working on things in the – essentially, 1000s B.C. – and what gets me is we always raise up the Greeks as the great architecture of modern culture, and so you have the Babylonians working in second millennium BC. You have the Egyptians working about the same time in second millennium BC. Flash forward to Classical Antiquity – by which I mean sometime after Christ. And, you have the Greeks getting into a philosophical argument – an existential – crisis over zero. Because, they didn’t want to say that there was something – and, by naming it, it became a thing.

Fraser: There was nothing.

Pamela: Yes.

Fraser: Right.

Pamela: Yes. So, you have this horrible mental stopping point in the years after Christ where they just basically had a mental breakdown over the, “But it’s a thing if we give it the name, but it’s a nothing.” For 2,000 years, no one had had issues in written documentation. Both the Babylonians and Egyptians appear to have gotten the concept of zero from other societies. This was not a new idea, but the Greeks had to have a mental breakdown over it because this is what the philosophical Classical Greeks did when you left them alone.

Fraser: Right, so they resisted.

Pamela: Yes.

Fraser: And, maybe considered the platonic ideal symbols for nothing – of nothingness.

Pamela: Ptolemy is one of the ones who actually wrote a lot on this. He was using a symbol for zero that he had borrowed from the Babylonians. He needed, in his mathematical works – which is how we ended up with the Babylonian sexidecimal system for basically all of astronomy. But, yes, the people in general – this was a problem.

Fraser: Okay. Who next? Because I think we’re moving towards – time-wise – the sort of more modern concept in the more modern society. So, what was another big development in zero?

Pamela: So, Fibonacci – who is by far one of my favorite – and, I feel like such a nerd having a favorite ancient person who worked in the sciences. Fibonacci – who is one of my favorite ancient mathematicians and scientists – He’s one of the ones who came up with the realization that we had all of these different mathematical series that describe all of these different spiral shapes, and everything. He’s starting to get into the post-Charlemagne era of Europe where it was now Italy, rather than Rome. And, he’s responsible for, basically, the word zero that we currently have.

Fraser: Really. I didn’t know that.

Pamela: So, it is stolen from other terms. What I particularly like is you may have heard the hot air balloons that were called Zephyrs, which comes from old roots of Zephyrus, which is the West Wind. And, the concept of zero is a bastardization of that idea of that nothing – that wind. And, it’s out of the Venusian way of speaking that this got turned into zero.

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And, Eddie has new toys to play with while the old ones are getting laundered. He has new treats to eat just as he runs out of treats from the prior month. Every month, there’s a little bit of joy, and you can see Eddie and his joy on our Twitter channel. So, check it out and see how much my dog loves Bark Box, and then get one for your dog, so your dog can love it, too. So, just to say it again, for a free extra month of Bark Box, visit BarkBox.com/astronomy and subscribe to a 6 or a 12 months plan.

Fraser: But, when did we really see that first zero – the shape?

Pamela: So, the first shape of zero – actually it goes back much further because if you look at it, and you look at the Greek concept of Theta, all of these were coming out at about the same time. Again, blame Ptolemy. Blame Ptolemy seems to be a fairly common thing to do.

Fraser: Don’t we already blame him for messing up the Solar System?

Pamela: Well, he was trying to do right by many things. He just didn’t always succeed.

Fraser: Yeah. But, I mean – and, I mentioned this earlier in the show – but, a lot of the mathematics that we do today – a lot of that came from a lot of the Arabic mathematicians – especially a lot of the best astronomy came out of that area, and they had a big hand in zero, too, right?

Pamela: One of the things that really was sort of brain-breaking that I came across while preparing for this episode – There’s a really cool book that is written from kind of like a first person passion project called Finding Zero: A Mathematician’s Odyssey. And, it’s one of those things that goes through and discusses all of the, “How do you track down the origins?” And, this includes looking on the sides of pyramids, and digging through jungles.

And, in the process of looking through the origins of numbers, it was realized that we often refer to our current numbering system as the Arabic/Hindi system, but Arabic numbers today, and Hindi numbers today look totally different. So, the symbols that we’re using – it’s a living system. Alphabets – the way we use them; the way they look – they all change over time. So, modern Arabic, ancient Arabic, the intellectual bias toward getting at the root of things has been fairly constant in the culture, but the way the numbers look have evolved, and I just think that’s so cool.

Fraser: Right.

Pamela: So, random rabbit hole. This is the episode of rabbit holes.

Fraser: No, no. [Inaudible] [00:19:44] rabbit hole. What was that book, again, that you mentioned? Finding Zero? What was it called?

Pamela: It’s Finding Zero: A Mathematician’s Odyssey to Uncover the Origins of Numbers. It’s by Amir Aczel. I may be pronouncing that horribly wrong. It exists both in Kindle – which is how I got it – and, there is also an audio version. I recommend the Kindle because there we symbols, and pictures, and diagrams, and I don’t want to think about what happened turning those into audio – and, I love audio books. Get this one on Kindle. Or, paper. Paper is a thing.

Fraser: So, when have we sort of stabilized on our modern thinking of zero?

Pamela: So, our modern ideas really started to percolate out during that Greek Classical period. And, while the philosophers were having existential crisis over giving a name to something that was nothing, the mathematicians were like, “We need this. We’re just going to use it, and move on.” And, by about the 5th century, it was just sort of an, “Okay. We’re good. We have nothing.” Literally, zero. They had nothing.

Fraser: Right. We had nothing. And, so then, from that point on, we’ve sort of been using that modern interpretation of zero. Now, you mentioned, sort of, earlier on, and we were talking about this before the show – that you were, sort of, running through the reasons we need zero, and we’ve, sort of, glossed over them a bit. We’ve talked about that you need a zero as a place-holder, but really what job is zero doing in our mathematics?

Pamela: So, mathematically, you need zero to be that thing that is between having something, and having a lack of something. And, the way to think about that is: On payday, I have something in my bank account, and hopefully a zero on my credit cards – in a perfect world that I don’t think has actually existed, but this is a goal. By the day before my next paycheck, I have significantly less in my bank account, and I have possibly and amount on my credit cards that indicates a total amount of less than zero. And, I’m just talking about day to day. I actually have retirement savings. No one worry about my future out there.

So, you need to be able to, say, when somebody owes you money, you need to have this concept of negative. When you have less than the average amount of water in a lake, you need to be able to have that concept of negative. And, to go from the negative numbers to the positive numbers, you have to have that point that is neither negative nor positive. And, zero is that point that is neither a something, or a lack of something. It is a nothing.

Fraser: Right. And, the key being that it is an integer like 1, 2, 3, 4, 5, and that it is that halfway point between 1 and -1 that as you’re counting from one to the other, if you don’t have that zero – if you just go from 1 to -1, your math is all messed up.

Pamela: And, this is really a concept that was worked on a great deal in the 7th century in India. India is another nation that has an amazingly rich history of advancing Algebra, advancing mathematics as a theoretical framework, as well as an applied subject. So, it is really amazing all of these different nations that we don’t learn about in school are actually the reason that we have science. It’s because they were advancing the maths.

Fraser: The three – did I miss one? There’s a place-holder, you need that halfway point between positive and negative –

Pamela: And, you also need to be able to denote that there’s a nothing. That thing that gave the Greeks an existential crisis.

Fraser: Right.

Pamela: So, you need a zero to say, “I currently have zero french fries.” This is a true statement.

Fraser: Yes. An existential crisis.

Pamela: Yes, yes. It is. And, to be able to express that lack of french fries, I need zero. Now, if I had twelve french fries, I don’t need a zero, but if instead I had skinny, little shoestring french fries cut into little pieces, and it turned out that I had 206 of them – because I was being a french fry eating crazy person, I need the zero as a place-holder. Now, if I had a small child who – and, I don’t – who was like, “Fries. Give me fries.” I could say, “I owe you a plate of 30 fries,” which requires a place-holder, “And, you will get it if you behave for the duration of this recording.”

So, say I’m speaking to an imaginary child who does not exist, I could have a deficit of 30 fries that I must somehow come up with in order to get said imaginary child to be quiet for the entirety of this episode. Again, I do not have said imaginary child.

Fraser: No, no. I understand. It’s possible that I have run those kinds of scenarios in reality.

Pamela: Yes. I believe this is where I learned these scenarios.

Fraser: So, one of the things that’s very interesting about zero – and, we deal with this all of the time in computers – is the divide by zero.

Pamela: Yes.

Fraser: That’s where zero really makes our lives miserable – when that zero ends up as the denominator of a fraction.

Pamela: And, this is something that is a new break-everything of the digital age, where you have to do numerical approximations to continuous functions. By which I mean, if you ever used Logo in school, you knew that the way you drew a circle was you moved 1 step forward; tilted 1 degree sideways. And, so you approximated drawing a circle by making a polygon of 360 sides. And, that is an approximation of a continuous function. An actual circle is a continuous function with infinitely small steps between each movement forward using that cosine and sine.

Now, with computers, they can’t handle continuous functions. They can’t handle the mathematical concept of infinity. And, so this is another place where – again, existential crisis related to a number – because how do you have a word for a thing you can never actually have, but yet we seem to actually – like, the idea of the universe being infinite is an idea. It could exist. The idea of there being infinite steps between me an my web-cam if I keep dividing the distance in half. Now, the reality is I’ll eventually bang my nose into my web-cam. But, computers can’t deal with this concept of infinity. They can deal with the concept of zero, and both are needed to have continuous functions.

Fraser: Right on. Alright. We’re out of time. Thanks, Pamela. We’ll talk to you next week.

Pamela: Okay. My pleasure.

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