Ramble, Part 48: Going Modular
Apr. 3rd, 2008 10:42 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe development of quaternions was only one of several more-or-less independent events in the rise of abstract algebra. As might be expected, Gauss's fingers were all over this as well.
For a couple of centuries, mathematicians had been investigating quadratic forms - polynomials in two or more variables, in which every term has total degree 2, such as x2+y2, or 2x2+xy-z2. Fermat, for example, had shown that a prime number can be written in the form x2+y2, where x and y are integers, if and only if it is one more than a multiple of 4. Others had considered Pell's equation, x2-Dy2=1, where D is some fixed positive integer. Both of these are special cases of the following: given a quadratic form, what integers can be written as values of that form, and if it is possible, in how many ways? Gauss, in his magisterial Disquisitiones Arithmeticae, introduced a novel method of studying such problems: modular arithmetic.
In its essence, modular arithmetic selects an integer - a "modulus" - and considers the effect of the operations of arithmetic on the remainders that are left on division by the modulus. For example, if the modulus is 8, we say that 3+5 is congruent modulo 8 to 0, because the remainder of 3+5, on division by 8, is 0. Likewise, 3x5 is congruent modulo 8 to 7. The significance of this, Gauss pointed out, is this. If a particular remainder modulo 8 cannot be represented by a given quadratic form (modulo 8), then no integer with that remainder can be represented by that form. For example, consider the form x2+y2 again. It is not hard to show that every square has remainder (modulo 8) 0, 1, or 4. Therefore the sum of two squares must have remainder 0, 1, 2, 4, or 5, and so no number with remainder 3, 6, or 7 can be the sum of two squares. By investigating the possible remainders modulo various different moduli, much information can be extracted, and the technique has proven extremely valuable in number theory.
For our purposes, what is significant about modular arithmetic is this. In speaking of, say, arithmetic modulo 7, it is certainly possible to treat it as referring to integers. However, the only number-symbols that are actually used are the remainders - 0 through 6 - and it is more efficient to regard the operations as actually involving, and yielding as results, those remainders: that is, saying such things as 3+5=1 and 4x4=2. But then what is it that these number-symbols refer to? Not, assuredly, to the integers that go by those names, since the sum of the integers 3 and 5 is not 1. Whatever these "integers modulo 7" are, they are not integers. Nor can they be fitted into the hierarchy of numbers (rational, real, complex, quaternion); they stand, somehow, outside. They are an entirely new kind of number.
It is not clear to me whether Gauss actually took this mental step. He does use such phrases as "the possible values of the expression square root of 2 modulo 7", which suggests at least an openness to this step. In any event, the step was taken soon after, and it became accepted that the proper realm of algebra included not only traditional numbers but also these new entities.
More were to follow.
Previous Next
Ramble Contents
For a couple of centuries, mathematicians had been investigating quadratic forms - polynomials in two or more variables, in which every term has total degree 2, such as x2+y2, or 2x2+xy-z2. Fermat, for example, had shown that a prime number can be written in the form x2+y2, where x and y are integers, if and only if it is one more than a multiple of 4. Others had considered Pell's equation, x2-Dy2=1, where D is some fixed positive integer. Both of these are special cases of the following: given a quadratic form, what integers can be written as values of that form, and if it is possible, in how many ways? Gauss, in his magisterial Disquisitiones Arithmeticae, introduced a novel method of studying such problems: modular arithmetic.
In its essence, modular arithmetic selects an integer - a "modulus" - and considers the effect of the operations of arithmetic on the remainders that are left on division by the modulus. For example, if the modulus is 8, we say that 3+5 is congruent modulo 8 to 0, because the remainder of 3+5, on division by 8, is 0. Likewise, 3x5 is congruent modulo 8 to 7. The significance of this, Gauss pointed out, is this. If a particular remainder modulo 8 cannot be represented by a given quadratic form (modulo 8), then no integer with that remainder can be represented by that form. For example, consider the form x2+y2 again. It is not hard to show that every square has remainder (modulo 8) 0, 1, or 4. Therefore the sum of two squares must have remainder 0, 1, 2, 4, or 5, and so no number with remainder 3, 6, or 7 can be the sum of two squares. By investigating the possible remainders modulo various different moduli, much information can be extracted, and the technique has proven extremely valuable in number theory.
For our purposes, what is significant about modular arithmetic is this. In speaking of, say, arithmetic modulo 7, it is certainly possible to treat it as referring to integers. However, the only number-symbols that are actually used are the remainders - 0 through 6 - and it is more efficient to regard the operations as actually involving, and yielding as results, those remainders: that is, saying such things as 3+5=1 and 4x4=2. But then what is it that these number-symbols refer to? Not, assuredly, to the integers that go by those names, since the sum of the integers 3 and 5 is not 1. Whatever these "integers modulo 7" are, they are not integers. Nor can they be fitted into the hierarchy of numbers (rational, real, complex, quaternion); they stand, somehow, outside. They are an entirely new kind of number.
It is not clear to me whether Gauss actually took this mental step. He does use such phrases as "the possible values of the expression square root of 2 modulo 7", which suggests at least an openness to this step. In any event, the step was taken soon after, and it became accepted that the proper realm of algebra included not only traditional numbers but also these new entities.
More were to follow.
Previous Next
Ramble Contents
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
no subject
Date: 2008-04-03 04:00 pm (UTC)The fault is all mine.
Are you writing a book? These posts could be drafts for making one.
Love, C.
no subject
Date: 2008-04-03 05:00 pm (UTC)As for the basic idea behind modular arithmetic: we actually use it in certain everyday situations, in particular in connection with time. If it's 8 o'clock now, what time will it be in 6 hours? Not 14 (=8+6), but 2 (the remainder of 14 after dividing by 12). Clock-time, in other words, operates (more or less) modulo 12. (Military time is modulo 24.) Or think of an odometer, as it reaches its limit and rolls back to all zeros.