 Our system of writing numbers is base 10 and positional. It's base 10 because each named unit, 1, 10, 100,000, and so on, is 10 times the magnitude of the proceeding, and positional because the value of a symbol is based on its location within the written number. A workable positional system requires a symbol indicating the absence of an order of magnitude. In other words, a symbol for zero. One problem with tracing the early history of our system is that there are two different zero symbols. The first is used to indicate the absence of a quantity. And in the 2nd century AD, Ptolemy used an open circle to indicate such absences, but Ptolemy's number system was not positional. Now, our modern zero symbol first appeared in the work of Pengala in 150 BC. But again, in Pengala's work, the symbol, which he refers to as sunya or void, is used as a marker without any numerical value. So when did our positional zero emerge? To answer that, it helps to consider a key problem before the invention of printing. How do you communicate an idea? One solution, poetry. So I'm going to mangle the Sanskrit pronunciation here, so bear with me. In Ptolemy, the Yavanajataka of Svujitvaja used an object numeral system to express numbers. In this system, a numerical amount was represented by an object that evoked the amount. For example, moon could mean one since there's one moon. Seriously? You're going to make me say it, aren't you? That's no moon. Well, okay, earth also could mean one. Eyes could mean two since most people have two eyes. Lim could mean six since the six parts of the Vedas are referred to as limbs. Teeth could mean 32 and so on. The value of a symbol depended on its place in the phrase with each position having ten times the value of the proceeding. For example, we might determine the value moon, eye, limb, moon. So we should read this as follows. Moon is one. Eye is two. But since it's in the second place, its value is ten times as great, so it represents twenty. Lim, remember, there are six limbs of the Vedas, that's six. But since it's a third word, it's ten times ten, a hundred times as great, six hundred. And again, moon is one. But as the fourth word, it's a thousand times as great, and represents one thousand. Now, it's important to note that the smallest order of magnitude is first. And so, more familiarly, we'd read this number as one thousand, six hundred, twenty, one. One intriguing feature is Spujit Vajra's use of words like void, or sky, or dot. And he's using these to correspond to the absence of an order of magnitude. Void, nothing, sky, wide open nothing. For example, moon, sky, eyes would be one, nothing, two hundred, or what we might call two hundred one. We see a variation of this approach in the work of Aryabhata, who replaced the object names with arbitrary consonants in his Aryabhatiya. Vowels could be interpolated to allow the word to be read. For example, the digits, six, three, three, three, five, seven, seven, five, could be associated with certain consonants. And to make the word readable, we'll interpolate some vowels, so we get the word kaya-gei-ni-su-khleur. And this would be, or as we would read it. And the clearest early evidence of a positional-based tense system comes from Ramagukta, who lived about a hundred years after Aryabhata. Translating from the Sanskrit, Ramagukta refers to the number four zeros, thirty-two, four, which would be zero, zero, zero, zero, thirty-two, four, which we would read as our modern number. Note that the thirty-two here is actually thirty-two ten-thousands, so four ends up at the millions place.