Zero

Back during the DonorsChoose fundraiser, I promised a donor that I’d write an article about the math of zero. I haven’t done it yet, because zero is actually a suprisingly deep subject, and I haven’t really had time to do the research to do it justice. But in light of the comment thread that got started around [this post][fspog] yesterday, I think it’s a good time to do it with whatever I’ve accumulated now.
History
———
We’ll start with a bit of history. Yes, there’s an actual history to zero!
In general, most early number systems didn’t have any concept of “zero”. Numbers, in early mathematical systems, were measurements of quantity. They were used to ask questions like “How much grain do we have stored away? If we eat this much now, will we have enough to plant crops next season?” A measurement of zero doesn’t really mean much; even when math is applied to measurements in modern math, leading zeros in a number – even if they’re *measured* – don’t count as significant digits in the measurement. (So if I’m measuring some rocks, and one weighs 99 grams, then that measurement has only two significant digits. If I use the same scale to weigh a very slightly larger rock, and it weighs 101 grams, then my measurement of the second rock has *three* significant digits. The leading zeros don’t count!) *(In the original version of this post, I managed to stupidly blow my explanation of significant digits, which several alert commenters pointed out. As usual, my thanks for the correction.)*
Aristotle is pretty typical of the reasoning behind why zero wasn’t part of most early number systems: he believed that zero was like infinity: an *idea* related to numbers, but not an actual number itself. After all, you can’t *have* 0 of anything; zero of something isn’t *any* something: you *don’t have* anything. And you can’t really *get* to zero as he understood it. Take any whole number, and divide into parts, you’ll eventually get a part of size “1”. You can get to any number by dividing something bigger. But not zero: zero, you can never get to by dividing things. You can spend eternity cutting numbers in half, and you’ll still never get to zero.
The first number system that we know of to have any notion of zero is the babylonians; but they still didn’t really quite treat it as a genuine number. They had a base-60 number system, and for digit-places that didn’t have a number, they left a space: the space was the zero. (They later adopted a placeholder that looked something like “//”.) It was never used *by itself*; it just kept the space open to show that there was nothing there. And if the last digit was zero, there was no indication. So, for example, 2 and 120 looked exactly the same – you needed to look at the context to see which it was.
The first real zero came from an Indian mathematician named Brahmagupta in the 7th century. He was quite a fascinating guy: he didn’t just invent zero, but arguably he also invented the idea of negative numbers and algebra! He was the first to use zero as a real number, and work out a set of algebraic rules about how zero, positive, and negative numbers worked. The formulation he worked out is very interesting; he allowed zero as a numerator or a denominator in a fraction.
From Brahmagupta, zero spread both east (to the Arabs) and west (to the Chinese and Vietnamese.) Europeans were just about the last to get it; they were so attached to their wonderful roman numerals that it took quite a while to penetrate: zero didn’t make the grade in Europe until about the 13th century, when Fibonacci (he of the series) translated the works of a Persian mathematican named al-Khwarizmi (from whose name sprung the word “algorithm” for a mathematical procedure). As a result, Europeans called the new number system “arabic”, and credited it to the arabs; but as I said above, the arabs didn’t create it; it originally came from India. (But the Arabic scholars, including the famous poet Omar Khayyam, are the ones who adopted Brahmagupta’s notions *and extended them* to include complex numbers.)
Why is zero strange?
———————-
Even now, when we recognize zero as a number, it’s an annoyingly difficult one. It’s neither positive nor negative; it’s neither prime nor compound. If you include it in the set of real numbers, then they’re not a group – even though the concept of group is built on multiplication! It’s not a unit; and it breaks the closure of real numbers in algebra. It’s a real obnoxious bugger in a lot of ways. One thing Aristotle was right about: zero is a kind of counterpart to infinity: a concept, not a quantity. But infinity, we can generally ignore in our daily lives. Zero, we’re stuck with.
Still, it’s there, and it’s a real, inescapable part of our entire concept of numbers. It’s just an oddball – the dividing line that breaks a lot of rules. But without it, a lot of rules fall apart. Addition isn’t a group without 0. Addition and subtraction aren’t closed without zero.
Our notation for numbers is also totally dependent on zero; and it’s hugely important to making a polynomial number system work. Try looking at the [algorithm for multiplying roman numerals][roman-mult] sometime!
Because of the strangeness of zero, people make a lot of mistakes involving it.
For example, based on that idea of zero and infinities as relatives, a lot of people believe that 1/0=infinity. It doesn’t. 1/0 doesn’t equal *anything*; it’s meaningless. You *can’t* divide by 0. The intuition behind this fact comes from the Aristotelean idea about zero: concept, not quantity. Division is a concept based on quantity: Asking “What is x divided by y” is asking “What quantity of stuff is the right size so that if I take Y of it, I’ll get X?”
So: what quantity of apples can I take 0 of to get 1 apple? The question makes no sense; and that’s exactly right: it *shouldn’t* make sense, because dividing by zero *is meaningless*.
There’s a cute little algebraic pun that can show that 1 = 2, which is based on hiding a division by zero.
1. Start with “x = y”
2. Multiply both sides by x: “x2 = xy”
3. Subtract “y2” from both sides: “”x2 – y2 = xy – y2
4. Factor: “(x+y)(x-y) = y(x-y)”
5. Divide both sides by the common factor “x-y”: “x + y = y”
6. Since x=y, we can substitute y for x: “y + y = y”
7. Simplify: “2y=y”
8. Divide both sides by y: “2 = 1”
The problem, of course, is step 5: x-y = 0, so step five is dividing by zero. Since that’s a meaningless thing to do, everything based on getting a meaningful result from that step is wrong – and so we get to “prove” false facts.
Anyway, if you’re interested in reading more, the best source of information that I’ve found is an online article called [“The Zero Saga”][saga]. It covers not just a bit of history and random chit-chat like this article, but a detailed presentation of everything you could ever want to know, from the linguistics of words meaning zero or nothing to cultural impacts of the concept, to detailed mathematical explanation of how zero fits into algebras and topologies.
[fspog]: http://scienceblogs.com/goodmath/2006/07/restudying_math_in_light_of_th.php
[saga]: http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM
[roman-mult]: http://www.phy6.org/outreach/edu/roman.htm

0 thoughts on “Zero

  1. Alejandro

    Nice post, I hope you continue it with an explanation of why x^0 = 1 for all x. In my experience non-mathematicians find that even more maddeningly unintuitive than the impossibility of dividing by 0.
    By the way, what is your source for the statement that Omar Khayyam and other Arabic mathematicians were acquainted with complex numbers? The accounts I have read credit Renaissance Italian mathematicians for first using them. Cardano is the name that comes to the mind.

    Reply
  2. Gabe

    Excellent. I’d love to see more posts about zero. It reminds me of my love for Xeno’s Dichotomy. I’m off to check out the Zero Saga.

    Reply
  3. Mark C. Chu-Carroll

    TomS:
    I said *arabic* scholars, not *arabian* scholars. The Persia of Khayyam’s time was definitely part of the arabic-speaking culture; Khayyam is often referred to as an Arabic scholar.

    Reply
  4. Davis

    If you include it in the set of real numbers, then they’re not a group…

    I’m going to be pedantic here, and note that you mean they’re not a group under the operation of multiplication. They’re certainly a group under addition.
    Just a friendly reminder that it’s always good form to state your group operation when there’s not a canonical choice…

    Reply
  5. Davis

    Nice post, I hope you continue it with an explanation of why x^0 = 1 for all x.

    I’ll give a couple of quick justifications.
    One property of exponents that should hold, morally, is that
    xm xn=xm+n.
    If you really want this to hold, you have no choice in defining x0:
    x0 xn = x0+n=xn,
    i.e., x0 has to be 1 (since one of the axioms of the real numbers is that the multiplicative identity is unique).
    Here’s another approach, via limits:
    Suppose k is a fixed positive number, and consider k1/n for n an integer (i.e., the nth root of k). As n gets larger, you’re taking a higher and higher root of k, which you can check with a calculator approaches 1 (regardless of whether k is bigger or smaller than 1, interestingly).
    In the limit as n approaches infinity, the exponent goes to zero, and the value of k1/n goes to 1. Since the function f(x)=kx is one we want to be continuous, you’re again forced to define k0=1.
    Personally, I find it much more weird to consider values of the form xr when r is an irrational number.

    Reply
  6. RPM

    It would be cool if colleges offered history of math courses. Do they? You could probably cover most of the material with only a solid understanding of algebra, right? Maybe a little bit of calculus for the more recent stuff.

    Reply
  7. Chris Nelson

    Am I missing someting or should “2 and 120 looked exactly the same” be “2 and 20 looked exactly the same”?

    Reply
  8. Davis

    I think most math departments offer history of math (I’m still kicking myself for not taking it as an undergrad), but usually the prerequisites entail a somewhat sophisticated background.

    Reply
  9. Davis

    Chris, the Babylonians used base 60. So a 2 in the second place means 2*60 (and a 2 in the third place means 2*602=2*720, etc.).

    Reply
  10. StephL

    Hey, this is a good thread. In fact, there’s a fantastic book written on the subject that is called Zero: The biography of a dangerous idea, and it is by Charles Seife. It also serves as an excellent introduction to ideas of limits and hence Calculus. I long ago decided that should I ever become a Calculus teacher I’d make this summer reading =).

    Reply
  11. Mark C. Chu-Carroll

    For why x0 = 1…
    Think about what exponents mean. One of the fundamentals that springs from their meaning is:
    xm×xn = xm+n
    So:
    xm×x0 = xm + 0 = xm.
    So x0 = 1.

    Reply
  12. Alejandro

    Just to make clear, I was not asking for an explanation for myself, just saying it would be nice to include such an explanation in a follop-up post for the sake of non-mathematicians.
    The question I was really interested in was the one about Arabic mathematicians and complex numbers…

    Reply
  13. Davis

    You guys are just *too fast*!

    Working on revising my thesis seems to motivate me to participate heavily in these comment threads. 🙂

    Reply
  14. Cody Cobb

    (So if I measure a weight, and it’s exactly 20 grams, my measurement has one significant digit; if it weighs 21 grams, my measurement has two significant digits. Zeros don’t count!)
    In my physics lab last semester, we would just underline the 0 if we meant it to be a significant digit and leave it bare if not.

    Reply
  15. EJ

    I usually stick to math by itself and don’t go in for that whole measurement business. This: “So if I measure a weight, and it’s exactly 20 grams, my measurement has one significant digit; if it weighs 21 grams, my measurement has two significant digits” is a surprise to me! I find that terribly counterintuitive.
    Could someone please define “significant digit” for me? That should fix me up.
    Thanks.

    Reply
  16. gk

    Awesome post. Is it possible to scrap Friday Pathological Programming (as I have never understood one post) for Friday Mathematics History? Or maybe make it Thursday.
    I have never heard or seen a history of math course, but most schools I have attended have a philosphy of math course which usually covers a lot of history.
    On that note, some philosophers, stubborn as they are, still denied the concepts of zero and infinity even up to the 19th century for the same reasons given in the post – that all concepts including numbers are based in the physical realm.

    Reply
  17. andrea

    A gal in our Entomology department had a great tee shirt=
    India’s contribution to mathematics: Zero
    (My other favorite tee of that semester was=
    Kansas an orogenic event in the making.
    For those of you not familiar with North American geography, Kansas is one of those flat rectangle states in the central US. As evidenced by the fossils, it was an inland sea 300 MYA, but no mountains to be had.)

    Reply
  18. Mark C. Chu-Carroll

    EJ:
    Significant digits are used in experimental physics. The idea behind it is that the precision with which you can calculate something based on experimental measurements is limited by the precision of the measurements. For example, if you measure something moving 14 meters on 2.1 seconds, it isn’t reasonable to say that you can calculate its velocity to be 6.666666666 meters/second. Your measurement’s were precise enough to be able to talk about the fifth digit of the velocity!
    The system that’s generally used is called significant digits. For each measurement, you count the number of digits that can have a meaningful effect on the computation. The overall result of the computation can only have as many significant digits as the *least* precise measurement that was used.
    The reason that 0 doesn’t count as a significant digit is because it’s *nothing*. A zero in a trailing position just eliminates the effect of that last digit of the measurement in a computation. 6.10 is no different in a computation that 6.1; the 0 doesn’t contribute any measurable effect in the computation, so it’s insignificant – its presence *has no effect*.

    Reply
  19. S. C. Hartman

    Having taught about “significant figures” to chemists for many years, I too was a bit puzzled about your statement that “zeros don’t count.” Converting to exponential (“scientific”) notation, 20 = 2.0 x 10^1, which is different from 2 x 10^1 when measuring a continuous variable. The former implies the measured value is between 19.5 and 20.5, while the latter implies the measured value is between 15 and 25. Counting the 0 in 20 as a significant figure has information for the precision of the measurement. In that sense, the zero does count.

    Reply
  20. EJ

    Mark and SC, thanks for your replies. I have to say that SC’s version makes more sense to me. I certainly agree that the 0 at the end of 6.10 makes no difference to a computation as far as the bare math goes. Hey, 6.10 = 6.1 period, end of story. But “significant figures” are about something other than the bare math. (I still don’t know how to write a formal definition, but I think I could make one up that is consistent with SC’s comment.)
    Hmmm… wikipedia also seems to be consistent with SC’s comment.
    http://en.wikipedia.org/wiki/Significant_figures

    Reply
  21. Polymath

    Thanks very much for the link to the process that the Romans used to multiply. My students have asked me this on occasion, and I never knew what to say. But it is especially relevant to this discussion because it shows the significance of the zero as a place-holder: the Romans did have a way multiply, but without zero, their method is independent of the notation (reading the linked page makes it clear that the method would work for any notation). What makes our way so much easier is that the zero allows notation-dependent multiplication.

    Reply
  22. Canuckistani

    On the topic of significant digits, I’d like to add that one of advantages of scientific notation is that it lets you unambiguously specify how many trailing zeros are significant.

    Reply
  23. Cat lover

    At the risk of opening up a big ol’ can of worms, I would like to point out that one *can* divide by zero… one just has to do it carefully! Consider the one-point (a.k.a projective) compactification of the real numbers, R^* := R U {oo}. (Note that we do not distinguish +oo from -oo. Distinguishing these leads to the two-point or affine compactification.) In this context, we do indeed have x/0 = oo for all nonzero x. Moreover, oo behaves more-or-less just like we expect it too: x*oo = x + oo = oo for all nonzero x, and x/oo = 0 for all finite x. Of course, we lose something along the way: the extended real numbers are no longer closed under addition, since oo – oo is undefined. We also cannot define 0*oo, 0/0 or oo/oo. Basically, this system is a rigorous way to get oo and 0 to behave as they “ought to” as much as possible.

    Reply
  24. Strabo

    That thing about 0 not counting for significant digits is different from what I’ve been taught.
    Consider I measure something and give the value as ‘6’ and something else as ‘6.000’
    I was taught that these meant different things, ‘6’ means that I’m confident that the value is somewhere between 5.5 and 6.5. And 6.000 means that I’m confident the value is somewhere between 5.9995 and 6.0005.
    Since ‘6.000’ is more precise, it seems to me that would we should say that I can give the value to more significant figures.

    Reply
  25. Mark C. Chu-Carroll

    Ok, so I blew it. It’s been more years than I want to admit since my last physics class, which was the last place I used sigfigs. I’d love to know just *where* I got that idea; I’m sure there’s something that I’m misremembering that I mangled into this idea about sigfigs. I’ll correct the post.
    -Mark

    Reply
  26. Zero

    Some of my thoughts on zero:
    O is a point. The shortest line possible is two adjacent points. (Zero’s axiom #1)
    O is one sign
    O is no sign
    O is nothing
    O = zero (no beginning and no end )
    O = ought
    O = nought
    O is gap filler “none” ” I looked for a man to fill the gap and found “none”.
    O is “no place” Luk 9:58 And Jesus said unto him, Foxes have holes, and birds of the air [have] nests; but the Son of man hath not where to lay [his] head.
    O is the one fold in infinity, eternity (size, direction, and time)
    O is the middle of each of your 6 senses.
    O is a reference point to begain all counting, both plus & minus
    O = 360 degrees
    O is a circle
    O = a minute, an hour, a day, a year ( a revolution )
    Here, O is real. Here in reality, it is the beginning. It is the inferred point at the end of your measuring stick. It is the bubble on a level. It is the middle of your bank account. It is the middle of every living thing, (fold). It is the middle of infinity. (size) It is the middle of eternity (time). It is the middle of direction…north & south, east & west It is the middle of each of your 6 senses.
    It is your, and my, middle face, indifference, between love and hate.
    Zero

    Reply
  27. Mark C. Chu-Carroll

    Alejandro:
    Unfortunately, I don’t know what source I got that from. I put together notes for this post as I’ve had time to research it over the last three weeks, but I didn’t write down what facts came from what source. The sources that I remember using include “The Saga of Zero” (as linked in the text), wikipedia (articles on zero, Brahmagupta, indian mathematics, arabic mathematics, arabic numerals), the sites of various math professors around the web, my old linear algebra textbook (out of print, alas, it’s a really great book!), Wolfram’s Mathworld, and an old dictionary of mathematics (McGraw Hill, I think). I looked at one or two other books on my office bookshelves, but I can’t remember which ones, and I’m at home at the moment. I *think* that that probably came from something about Brahmagupta; I remember that as I was reading about him, I was finding really fascinating links to all sorts of things.

    Reply
  28. Mark C. Chu-Carroll

    You know, I was wondering how long it would take before this post attracted some kind of crazy numerologist. Apparently the answer was “approximately 24 hours”.

    Reply
  29. Torbjörn Larsson

    A cheat sheet on sigfigs is to note the display of a multimeter – trailing zeros is just another digit.

    Reply
  30. keiths

    Sam Phillips has a song entitled “Zero Zero Zero!”:
    big numbers go by
    I close my eyes
    I never count a large amount
    my lucky number is below one
    you never know when you might need a zero
    the zero in my hand
    is nothing to lose
    it’s hard to confuse power with love
    love with power
    everything that I’m not is all that I’ve got

    Reply
  31. JuanCarlos

    Just to get back to the x0 = 1 issue.
    The way I got that knowledge was from exponent properties too:
    xm/xn = xm – n
    then,
    xm/xm = xm – m = x0
    but, on the other hand
    xm/xm = 1
    then,
    x0 = 1, for all x ≠ 0.
    So, 00 is also meaningless, right?

    Reply
  32. Canuckistani

    Actually, JuanCarlos, 00 is quite an interesting expression. Technically, it’s an indeterminate form, meaning that more information must be added before it can be given a particular value. But in a sense, 00 is almost always equal to one. The function f(x, y) = xy is discontinuous at (0, 0), but it approaches one along any path in the xy-plane that is bounded away from tangency to the y-axis.

    Reply
  33. Brandon

    JuanCarlos when you put it in that fractional form it becomes the indeterminate 0/0 which means it can have a solution.

    Reply
  34. Xanthir

    A quick primer on sigdigits. ^_^ From my high-school physics class.
    First, a mnemonic: Absent, atlantic; present, pacific. If you’re in the Americas, this tells you that if there is a decimal point, you count sigdigs from the left to right. If there isn’t, you count from right to left.
    Once you know which direction to go, you simply start counting at the first non-zero digit, and then keep counting until the end (this includes any zeros that might be present).
    So, 200 has 1 sigdig. 200.0 has 4. If you want to express 200 to 2 sigdigs, you have to use scientific notation and say 2.0 x 10^2.
    When multiplying, the answer should have as many sigdigs as the number with the greatest sigdigs. If you multiply 200 by 4.55, the answer must have 3 sigdigs, and thus is 9.10 x 10^2, not 910). When adding, you do the reverse, and use the least number of sigdigs. 200 + 4.55 has 1 sigdig in the answer, and so is simply 200.
    In chemistry, we usually just say everything uses 3 sigdigs no matter what. This isn’t experimental chem, though, just ordinary college experiments meant to showcase a point.

    Reply
  35. DouglasG

    My alma mater (when I went there) had “History of Mathematics” courses. University of Northern Iowa. Enjoyable classes for sure!
    By the way, the Mayan’s also had a zero in their number system.
    Significant digit stuff is for those applied math types. So, it isn’t important…

    Reply
  36. Zero

    You know, I was wondering how long it would take before this post attracted some kind of crazy numerologist. Apparently the answer was “approximately 24 hours”.
    Posted by: Mark C. Chu-Carroll | July 22, 2006 05:08 PM
    —————————————————————-
    Mark, Andrea and I were talking about zero two days before you posted “Zero”
    —————————————————————–.
    andrea:
    I think of zero and infinity as being like obverse and reverse of the same coin.
    Andrea, there are,IMO, two infinities. Infinity small (-)
    and infinity big (+). Zero is in the middle.
    Posted by: Zero | July 20, 2006 10:35 PM
    ——————————————————————
    I was disappointed when your only response to my post was name-calling.
    My biggest surprise however, was that no one came to my defense, whether my math was good or bad.
    Jesus (74) + O (nothing in me) = God (704)
    Jhn 14:30 Hereafter I will not talk much with you: for the prince of this world cometh, and hath nothing in me.
    Jesus folded = fold (37)
    Psa 37:37 Mark the perfect [man], and behold the upright: for the end of [that] man [is] peace.
    Rev 13:18 Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number [is] Six hundred threescore [and] six.
    Jesus x 9 = 666
    Jesus + 9 = wisdom
    Apparently if there is a God, he is a crazy numerologist also. I’m in good company.
    Zero

    Reply
  37. Bronze Dog

    What other response is there, Zero?
    How does 74 + 0 = 704? Last time I checked, 74 + 0 = 74.
    As for your name play with Jesus, you’ll have to explain what your assumptions are, and why we should accept them.

    Reply
  38. Mark C. Chu-Carroll

    Zero:
    I’ve made it abundantly clear in other posts on this blog just what I think about numerology in general; and what I think about half-assed numerology in particular. Just click on the “numerology” link in the categories.
    Playing with english letters as numbers, you’ll find patterns wherever you go looking for them. We humans are really good at finding patterns, even when there aren’t any.
    The fact that your numberplay doesn’t even work without all manner of trickery just makes it that much sillier. (I mean really, you couldn’t make it work with just addition, so you needed to do an addition by lexical insertion to try to connect jesus, 0, and god? I’m supposed to take this seriously, even as numerology?)

    Reply
  39. JuanCarlos

    Thanks Canuckistani and Brandon.
    For a moment I fell in the common confusion of the difference between 1/0 and 0/0, just the first one is meaningless, right?
    Now, taking into accout your comments I have two questions: could 00 = 1 be considered as a convention?
    and could 0/0 be interpreted as a notation for all the whole numbers?

    Reply
  40. Xanthir

    Properly, they’re both undefined. 0/0 is one of the classical undefined quantities, but it’s essentially the same as 1/0 (someone correct me if I’m wrong). Both have no computable value without adding more information into the system.
    0^0 is also undefined. I can come up with reasonable arguments for it being 0 or 1. Thus, I can come up with arguments for it being *anything*.

    Reply
  41. JuanCarlos

    Well, I notice a difference between 1/0 and 0/0 if I consider the fraction as a division and think that n/m denote a number x that satisfies x * m = n.
    In the 1/0 case you are searching for a number x that satisfies x * 0 = 1, which doesn’t exist. However, in the 0/0 case, you are searching for a number x that satisfies x * 0 = 0, which could be any number, that’s the reason I asked if 0/0 could be considered as a notation for all the whole numbers.

    Reply
  42. Zero

    Mark, I did read all of your numerology links and I agree… they make no sense.
    I use the most simple code possible, a numerical value for each letter of the alphabet:
    A equals 1. Z equals 26. I total up the individual letters in a word to get its value. An example:
    point (16+15+9+14 +20) = 74 (GD).
    Jesus = 74
    messiah = 74
    yshua = 74
    cross = 74
    beauty = 74
    heavens = 74 (the heavens declare)
    clouds = 74 (coming in the clouds)
    signet = 74
    between = 74 (74/2 = 37) (cg) (fold)
    gospel = 74
    finished=74 (It is finished. )
    zero = 64
    Israel = 64
    Zion = 64
    dust = 64 ( 64 % of 7000 = 4480 )
    feet = 36 ( 36% of 7000 = 2520 ) seven circles
    seven fold (128) divided by 2 = six fold (64)
    love = 54
    banner = 54 (His banner over me was love.)
    roof = 54 (roof + 12 = floor)
    eyes =54 ( Through the eyes of love.)
    Our = 54 ( O you are.) (The first word in the Lord’s prayer.)
    equals = 54
    I have, in my study of numbers, 4 values for “God”. 26, 54, 704, & 1260.
    I Am + I Am + I Am = 23+23+23 (Jehovah) (WWW)
    Did you hear about the spider sooo big he had a world wide web?
    God x God x God ……………………………… = 47,900,160
    704 x 54 x 1260……………………………… = 47,900,160
    70 seven fold x 99 x love…………………. = 47,900,160
    God x 1260 x love ……………………………. = 47,900,160
    10 % of heaven seven fold x God x God = 47,900,160 (heaven = I Am + Eve) (23 +32)
    10 % of 1x2x3x4x5x6x7x8x9x10x11x12….= 47,900,160
    God’s reward for healing 10 lepers…one “Thank you.”… 10 %.
    iron (9 x 18 x 15 x 14) x 1408 (2 x God)= 47,900,160 Rev 12:5
    Jhn 1:1 In the beginning was the Word, and the Word was with God, and the Word was God.
    God x G x O x D x (God + God + God )…..= 47,900,160
    Word (22,680) x 2112 ……………………….. = 47,900,160
    Jesus said, ” I have chosen you 12.”
    1+2+3+4+5+6+7+8+9+10+11+12…………….. = 3 x God ( 78)
    13+14+15+16+17+18+19+20+21+22+23+24= 3 x Jesus (222)
    704 x 3 = 2112 (U 12)
    47,900,160 minus (Jesus squared x 135) = 47,160,900
    One divided by Jesus (74 ) = .0135135135135…( ace to infinity )
    word = God (54) x G x 0 x d (7 x 15 x 4)
    word = 54 x 7 x 15 x 4
    word = 22,680
    word = 2/3 of iron ( 9 x18 x 15 x14) (Rev. 12:5)
    Rev 22:13 I am Alpha and Omega, the beginning and the end, the first and the last.
    I is the first letter in the bible.
    N is the second, and also the last letter in the bible.
    I x N = 126 (AZ)
    word = 126 squared + (126 x love )
    word = First ( I ) x Last (N) x 360 over 2
    word = First x Last x 180 (AZ x fold) Zec 9:9
    180 = First (9) x Last (14) + love (54)
    word = one (ace) x 1 week (168 hours)
    word = 90 % of seventy 360’s (the remainder is two 1260’s)
    word = first x last x it (I Am x it )
    First x Last ( 126 ) + love (54) = 180
    word = 126 (az) x 180 (9 x 20)
    Jhn 1:1 In the beginning was the Word, and the Word was with God, and the Word was God.
    word = God (54) x G x 0 x d (7 x 15 x 4)
    First x last (9 x 14) = Alpha & Omega (126) Beginning & End (AZ)
    First (9) + Last (14) = I Am (23) ( 3 x I AM = Jehovah )
    (First x Last) + love = 180 (Man’s best friend is God turned backward.)
    First x Last x 180 = Word
    First x Last x 180 x (3 x God ) = 47,900,160
    First + Last + love = 77 (Christ)
    God = seven 360’s folded (1260)
    God = 704 (beginning, end & nothing) (Jesus + nothing in me) Jhn 14:30 Hereafter I will not talk much with you: for the prince of this world cometh, and hath nothing in me.
    God = 54 (first, last,& nothing) (Ass + 15) Zec 9:9
    God = 26 ( a through z ) also G + o + d (7 + 15 + 4) God + 12 = Gold , Noah
    God = U 12 (2112) divided by 3 (11 a woman hid in 3 measures)
    God = 11 hidden in 3 measures (G , O , & D )
    God = Eve x 22 (704)
    God = Eve + 22 (54)
    God = 10 % of heaven sevenfold (10 % of 55 x 128 )
    God = the total of the first 12 #s divided by 3 (1+2+3+4+5+6+7+8+9+10+11+12 divided by 3)
    Jesus = the total of the second 12 #s divided by 3
    ( 3 x God ) + (3 x Jesus ) = the total of the first 24 #s
    666 = the first 12 + the second 12 + the third 12 (check out a roulette table ) the total of the 4 center numbers on the table, 14, 17, 20, and 23 = 74, Jesus (Gd).
    666 = 9 x Jesus
    666 + 12 = 378 (3 x God ) + (3 x Jesus ) + ( 3 x Alpha and Omega or 3 x the beginning and the end or 3 x the first and the last )
    Jesus x 12 = 888
    God (704 ) = 88 x 8
    Jesus x 24 = 1776 Rev 4:4And round about the throne [were] four and twenty seats: and upon the seats I saw four and twenty elders sitting, clothed in white raiment; and they had on their heads crowns of gold.
    historical date………7/04/1776
    In summary, life is not about numbers. It is about family and home.
    IMO, chos is natural. Order is mind made.
    It’s easer to form order from chaos than to create something from nothing. (See Gen. 1:2)
    Blessings
    Zero

    Reply
  43. Xanthir

    Zero: You’re crazy. Mark’s explained a couple of times in the past why numerology is no good. One of the most important things is that you’re doing it all in English. To the best of my memory, Jesus never spoke English, and the disciples certainly didn’t write in it. And you’re still twisting things to conform to your expectations. You misspelled Yeshua, for example. How many links occur in Hebrew? Greek? Spanish? Russian? If I can derive numerological ‘truths’ in these languages that contradict what you find in english, what does that say?
    JuanCarlos: Hmm. Well, oo * 0 can equal 1. It’s another classical undefined quantity, after all. ^_^ Point taken, though.
    0/0, while it can potentially be any number, isn’t every number. For example, if a function yields 0/0, by L’Hopital’s rule you take the derivative of the numerator and denominator until you get a defined quantity, and this is the value of the original expression. In other words, it’s not that 0/0 is every number, it’s that 0/0 is *no* definite number. It hides the true value of the expression.

    Reply
  44. Davis

    In other words, it’s not that 0/0 is every number, it’s that 0/0 is *no* definite number. It hides the true value of the expression.

    Properly speaking, an expression with 0 in the denominator is not defined, period — there’s no “true value.” When you use, say, L’Hopital’s rule to determine the limit of such an expression, you’re figuring out what the value of the expression should be in order to assure continuity. The same thing goes with the other indeterminate forms.
    This may seem like a subtle distinction, but in general there’s no reason to assume that functions are always defined at such points so that they are continuous.

    Reply
  45. Mark C. Chu-Carroll

    Zero:
    You’re quite amusing. Even as you try to defend yourself, you’re demonstrating what’s so damned silly about the kind of numerology you’re doing.
    You start by saying that you take the english letters, give each one a number, and then just add things up. But then, you use words with letters omitted (Yshua instead of Yeshua); you add numbers up wrong (equals does *not* add up to 54); you throw in multiplications, divisions, and roots at random. If you’re allowed to play with numbers any way you want, you can always find *some* sequence of arithmetic operations that give you the result that you want to find.
    And that’s igoring the fact that you’re doing it all in english; the supposed holy texts that all of this comes from were written in (variously) hebrew, aramaic, or greek, and translated into english, often through an intermediate language. If you allow yourself the additional freedom of translation – which allows you to pick and choose words – then you get even more freedom to create a pre-specified result.

    Reply
  46. Xanthir

    Davis: Thanks for the clarification. I knew what you said, but I’d glossed over it in my remembering. We often want to treat functions as everywhere continuous because it’s easier that way, and L’Hopital’s rule is one way to do so. But you’re right, if a function yields 0/0 at some point it is well and proper undefined at that point. Any rules that give a real value to it simply smooth the function out to eliminate the problem.
    Of course, most of the time that’s exactly what we want. If we’re looking at a function that’s supposed to be modelling the real world, discontinuities aren’t really very useful. The real world is continuous (except perhaps in certain pathological instances), and so smoothing our model over gives us the value we’re actually looking for.
    But as I said, point taken. ^_^

    Reply
  47. Davis

    Xanthir, I always try to be picky about these things because I find pathological function behavior fun. If you insist on things being “nice,” you miss out. 🙂
    My favorite example to bring up is that “most” continuous functions are nowhere differentiable, which is extremely un-nice. But this type of function seems to be a good way of modeling Brownian motion.
    (Apologies for getting way off topic.)

    Reply
  48. Zero

    Mark, you are right. I did, in haste, err. “equals” does not = 54. However, as you know,
    if you scribe the path of a circumference unpealing from the top of a circle for 360 degrees, then
    continue from that point and wrap 360, it forms a perfect heart. (Or maybe a leaf)
    Therefore, in my crazy way of thinking, being tied to an apron string is not a bad thing. Luk 15:11
    Zec 9:9 Rejoice greatly, O daughter of Zion; shout, O daughter of Jerusalem: behold, thy King cometh unto thee: he [is] just, and having salvation; lowly, and riding upon an ass, and upon a colt the foal of an ass.
    Mar 10:9 What therefore God hath joined together, let not man put asunder.
    (AZ under) my words
    What God = 78 (3 x God)
    joined together = 222 ( 3 x Jesus )
    25 through 36 + 12 = 3 x ( God + Jesus + God ) (378) (AZ x 3)(IxNx3) (First x Last x 3)
    ass (39) + O (15) = 54 (love)
    351(A through Z) + 135(ace) = 360 + 126 (54/40) (39/9 + 15/9)
    I forgot to add “fruit” = 74 in my prior post.
    Rev 22:2 In the midst of the street of it, and on either side of the river, [was there] the tree of life, which bare twelve [manner of] fruits, [and] yielded her fruit every month: and the leaves of the tree [were] for the healing of the nations.
    Xanthir, I flunked Spanish 101 in my one year of college and if God doesn’t speak English,
    I’m in real trouble.
    I asked my English golfing friend if they observed the 4th of july, since we won the war.
    He replied, ” Hell no!”
    I asked, “Then how do you get from the 3rd to the 5th.”
    Web Results
    Page 1 of 773 results containing yshua (0.07 seconds)
    Blessings
    Zero

    Reply
  49. N@

    Wow, all these comments, and only one mention of the Maya? The Maya had the zero before the Hindus (and for lack of a better source, Wikipedia agrees). But I understand for the purposes of your entry, their notion of zero was not communicated to Europeans the way Arabic numerals propagated; and your site is (again, understandably) primarily concerned with European mathematics. But no post on zero would be complete without mentioning the impressive Mayan system.

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  50. Xan

    From Brahmagupta, zero spread both east (to the Arabs) and west (to the Chinese and Vietnamese.)

    Erratum: “east” and “west” appear to be swapped.
    Delete this post when the text is fixed.

    Reply
  51. Anonymous

    So, for example, 2 and 120 looked exactly the same – you needed to look at the context to see which it was.
    should be
    So, for example, 12 and 120 looked exactly the same – you needed to look at the context to see which it was.

    Reply
  52. Jonathan Vos Post

    Fibonacci showed that the solution to the cubic equation
    x^3 + 2x^2 + 10x = 20
    can have no solution of the form a + sqrt(b) , where a and b are rational. He gives an approximation:
    1; 22, 7, 42, 33, 4, 40
    best to that time, and for another 300 years. Note the use of sexigesimal numbers.
    That approximation, converted to a decimal, is accurate to 8 digits! We do NOT know what his method was for finding this root.
    http://www.math.tamu.edu/~don.allen/history/mideval/mideval.html

    Reply
  53. Nathanael Nerode

    Why is x^0 defined to be 1 for all x? Because it makes a lot of things work out very tidily.
    Exponentiation x^y is defined on numbers other than positive integers by making a new “extended” definition which agrees with the basic definition of exponentiation (as repeated multiplications) for the positive integers. An extended definition must be suitable for the purposes we are going to use exponentiation for, and the extensions which are “standard” seem to be suitable most of the time they are used. Unfortunately the extensions are not quite all compatible with each other. There are two sets of extensions:
    (1) Extensions based on a positive real-valued x (we can extend the definition to any complex y in a way which makes f(y)=x^y a smooth function, but it doesn’t work when x isn’t a positive real). This is what gives you x^0=1. Extending to x==0 gives you 0^0=1.
    (2) Extensions for zero, negative real, and complex values of x (this is based on the repeated multiplication definition, and only really works for positive and negative integer values of y). Extending to y==0 gives you x^0=1 for most x, based on multiplication principles, but gives you 0^0=0.
    0^0 comes out differently from the two extension methods. Luckily it’s not a useful value to define most of the time, so we don’t.
    BTW, you can coherently define an extended number system where 1/0=Infinity, and it’s actually rather useful for some purposes.
    It’s actually very important to specify your domain when you’re talking about any mathematical operation. Sometimes exponentiation with real numbers makes sense for your problem. Sometimes it does *not* make sense and you only want integer exponents.

    Reply
  54. Xanthir, FCD

    Yep, that’s akin to the argument that Polymath uses. 00 can arguably be defined as either 0 or 1 based on common trends.
    His way of explaining it, though, might have made a bit more sense. He basically drew a table, where each entry was XY. Going horizontally (where Y is held constant), we find that every value of the form X0 equals 1. Going vertically, we find that every value 0Y equals 0.
    His argument was that both trends break when you reach 0n<0, becoming undefined. As well, at 00, one of the trends has to break. Since the vertical line (0Y) goes completely undefined one step later, one might as well just break it here, and allow the horizontal line (X0) to stay constant, and so he defines 00 as 1.

    Reply

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