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## Hot answers tagged number-systems

179

Short answer: your confusion about whether ten is special may come from reading aloud "Every base is base 10" as "Every base is base ten" — this is wrong; not every base is base ten, only base ten is base ten. It is a joke that works better in writing. If you want to read it aloud, you should read it as "Every base is base one-zero". You must ...

65

Many people believe that since humans have $10$ fingers, we use base $10$. Let's assume that the Martians have $b$ fingers and thus use a base $b$ numbering system, where $b \neq 10$ (note that we can't have $b=10$, since in base $10$, $x=8$ shouldn't be a solution). Then since the $50$ and $125$ in the equation are actually in base $b$, converting them to ...

48

Many languages have (at least relicts of) non-decimal counting, very often vigesimal (because we have 20 fingers plus toes), but also many other systems. I recommend an old Gutenberg project of mine, The Number Concept Note for example that the Danish word for 55 is femoghalvtreds "five more than half the third twenty-block"

32

Alas, there are no algebraically coherent "triplexes". The next step in the construction as has been said already are "quaternions" with 4 dimensions. Many young aspiring mathematicians have tried to find them since Hamilton in the 19th century. This impossibility links geometric dimensionality, fundamental properties of polynomial equations, algebraic ...

31

You're exactly right that such a system would be represented by the use of arbitrary tally marks. Such a system is known as a Unary Numeral System (Wikipedia Entry): The unary numeral system is the bijective base-1 numeral system. It is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen ...

26

13 fingers. Translate $5x^2-50x+125$ into base-$b$: $$5x^2-(5b)x+(b^2+2b+5)$$ Since this has roots $x=5$ and $x=8$ we must have $$5x^2-(5b)x+(b^2+2b+5)=k(x-5)(x-8)=kx^2-13kx+40k$$ so, equating coefficients, $$5=k,\quad 5b=13k,\quad b^2+2b+5=40k$$ and so $b=13$. It's easy to check that the last equation is satisfied as well. Perhaps the Martians had ...

26

actually numbers from 11 to 16 are quite regular in French (and in Italian) too: they just are a derivation from Latin. | French | Italian | Latin un | on·ze | un·dici | un·decim deux | dou·ze | do·dici | duo·decim trois | trei·ze | tre·dici | tre·decim quatre | quator·ze | quattor·dici | ...

23

Actually, if you go back in time a bit in English, you'll realise that English was 'strange' too: Four score and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty, and dedicated to the proposition that all men are created equal. (The Gettysberg Address, 1863) Now if you were to translate that into French in ...

21

If we are working in base $b$ (we must have $b\gt3$), then $0.3333\ldots$ is $$0.3333\ldots = \frac{3}{b} + \frac{3}{b^2} + \frac{3}{b^3}+\cdots$$ Since $$\sum_{n=1}^{\infty}\frac{3}{b^n} = \frac{3}{b}\sum_{n=0}^{\infty}\frac{1}{b^n} = \frac{3}{b}\left(\frac{1}{1-\frac{1}{b}}\right) =\frac{3}{b-1},$$ then...

20

In an interview, you can impress the interviewer, by mentally calculating and determining the result. As other answers have mentioned, you need to express the equation in base different from 10 and then equate it with the roots of the equation. From the options, its clear that the base is greater than 10. That means $5$ and $8$ are unit digits in some ...

20

Of course it seems natural to you; you grew up in the modern world, where everyone accepts zero. More importantly, people now accept the abstract concept of numbers and are capable of divorcing them from the things that they represent. This is a sophisticated point of view. From a more naive point of view, a number is a property of a collection of ...

19

As you mentioned, $$6 = {\color{red}1}\cdot 2^2+ {\color{red}1}\cdot 2^1+{\color{red}0}\cdot 2^0 = {\color{red}{110}}_B.$$ Analogously $$\frac{1}{4} = \frac{1}{2^2} = {\color{red}0}\cdot2^0 + {\color{red}0}\cdot 2^{-1} + {\color{red}1}\cdot 2^{-2} = {\color{red}{0.01}}_B.$$ Edit: These pictures might give you some more intuition ;-) Here $\frac{5}{16} = ... 19 Every finite-dimensional division algebra over$\mathbb{R}$is one of$\mathbb{R}$,$\mathbb{C}$or$\mathbb{H}$. This is what is called the Frobenius Theorem. You may refer to here for details. 18 The natural numbers can be defined by Peano's Axioms (sometimes called the Peano Postulates): Zero is a number. If n is a number, the successor of n is a number. zero is not the successor of a number. Two numbers of which the successors are equal are themselves equal. (induction axiom.) If a set S of numbers contains zero and also the successor of every ... 18 Expanding on the comment by J.M., let me quote from the (highly recommended) book by Georges Ifrah The Universal History of Numbers (Wiley, 2000, pp. 21-22): Traces of the anthropomorphic origin of counting systems can be found in many languages. In the Ali language (Central Africa), for example, "five" and "ten" are respectively moro and mbouna: moro ... 18 The existence of a bijection between the class of ordinals$On$and the class of surreal numbers$No$is independent of the axioms of set theory. There are several interesting possibilities: If ZFC is consistent, then there is a model of ZFC in which there is a definable such bijection. This is true in Goedel's constructible universe$L$, for example, for ... 18 I would like to expand on Trevor Wilson's answer. Base-$b$representation of integers is rooted in the fact that, for any non-negative integer$n$, there is a unique representation of$n$in the form $$n = \sum_{i=0}^\infty a_ib^i$$ where$0 \le a_i < b$. For example, when$b$is 3, and$n$is 47, the unique solution has$a_0 = 2, a_1 = 0, a_2 = 2, a_3 ...

16

What you're suggesting is known as binary coded decimal representation. It is used by some simple calculators, and used to be commonly used in financial computing, because it can represent dollar amounts without rounding the cents (and also makes it easier to produce decimal output for human consumption). But it is harder to calculate with than true binary ...

16

There are two terminologies that I'm familiar with. Sometimes, the part to the right of the decimal (cents) is called the mantissa, and the part to the left (dollars, in your metaphor), is called the characteristic. But I also like the generic terms integer-part and fractional-part. It's what I and those with whom I do research call them (who uses the work ...

15

I think the answer here might be, that the guys who thought base 10 was a good idea had the largest sticks. If one trusts the wikipedia, the Babylonians had a base 60 system, which can still be felt today with this "60 minutes in an hour" nonesense, and a (related) base 12 system was widely in use too. There are still unique words for "eleven" and "twelve", ...

15

Yes, ten ( ..... ..... ) is a special number. Not magical but special because it a very convenient base for species that have ten fingers. Arguably we can use hands and fingers to encode 1024 numbers using the binary system, but that would be less robust across reading directions and some configurations/gestures are physiologically hard to do.

15

Apparently you were looking for the taxicab numbers whose name derives from an anecdote of G.H. Hardy on his visiting Ramanujan: I remember once going to see him [Ramanujan] when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he ...

13

I think you were being a little too hard on Isaac. The truth is that the real numbers are a sophisticated mathematical construction and that any explanation of what they "are" which pretends otherwise is a convenient fiction. Mathematicians need these kind of sophisticated constructions because they are what is required for rigorous proofs. Before people ...

12

Actually, the values of the letters continue up to 90, then 100, 200, 300, and finally 400 for the tav (see the Wikipedia page on Gematria). (The values for the "terminal letters", like the nun-sofit, are, I believe, more "recent"). But this is not a base system, because the system is not positional, it is aggregate: the value of a latter/symbol does not ...

12

The quaternions and octonions are number systems that extend the complex numbers. Together with the complex numbers and the real numbers themselves, these form the only normed division algebras over the real numbers. In a normed division algebra, there is a notion of "size" (given by the norm), and every non-zero element has a left and right multiplicative ...

12

You are misinterpreting the statement. "All primes satisfy property $X$" means "If $p$ is prime, then $p$ has property $X$." You have instead interpreted it as "If $p$ has property $X$, then $p$ is prime." The statement is true, because if $p$ is a prime greater than $3$, then $p$ is not divisible by $2$ or $3$, whereas a number whose base six expansion ...

12

I don't know where you read that a number system can have matrices as a base. I did come across what follows, but I don't think the claim is that "any" ("every") number system can have matrices as its base, nor that if a number system exists with all matrices serving as its base; rather, if such a base exists, the criteria for determining the matrices that ...

12

I will give a point which was amiss in both the answers and somewhat connects this question to the set theoretic tags it has. There can be a largest number system, in the sense of ordered fields (that is it embeds $\mathbb R$ but not $\mathbb C$) and that is The Surreal Numbers. It is a class field, which means it is not a set and has no cardinality. As an ...

11

Vhailor's answer takes care of most of your questions. I'll try to help out with the rest. I'm not sure what it means for a mathematical concept to have a purpose, but I would say the purpose of $\mathbb{H}$ and $\mathbb{O}$ is that, together with $\mathbb{C}$ and $\mathbb{R}$ itself, they are the only finite dimensional normed division algebras over ...

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