We have somehow avoided talking about perhaps the most fascinating number of all, the square root of -1, frequently called simply 'i'. You may have run into it in connection with the quadratic formula, which sometimes presents you with the square root of a negative number. Historically such numbers were first seriously discussed by Italian algebraists in the 16th century. A formula for the roots of a cubic polynomial equation, now called Cardano's formula -- a very complicated and mysterious formula! -- occasionally produces correct results only if you allow individual terms in the formula to have these "imaginary" parts. That seemed to suggest that there was value in regarding i as a genuine number, to be manipulated just like other numbers, with the understanding that i^2=-1. Stories of this period in mathematics include zany and colorful characters, not the least of them being Cardano himself. The book in which his formula first appeared, Ars Magna (1545) is available in a modern edition in English translation, and is in our library. You will get some feeling for the period if you look at it. You can also find some of this material on the web, including capsule biographies of Scipione del Ferro, apparently the original discoverer of the cubic formula, Nicolo Tartaglia, who discovered it independently, and Ludovico Ferrari, Cardano's brilliant student. No real use of the imaginary number 'i' was made in this period, but it was noted: Cardano, after trying unsuccessfully to make sense of it, said it was "as subtle as it is useless."
I will not attempt to describe all this on a web page, but just limit myself to the elementary rules of complex arithmetic in a few examples, which will not seem so subtle, I hope. Here is an example of adding two complex numbers (numbers of the form a+b*i):
(2+3i)+(4+i)=6+4i
An example of multiplying:
(2+3i)*(4+i)=8+12i+2i+3i^2=8+14i-3=5+14i
Notice how you multiply complex numbers like binomial expressions (a+b)*(c+d)=ac+ad+bc+bd. If b and d each contain the factor 'i', then bd contains two factors of 'i', i.e., (-1).
An important special case:
(a+bi)*(a-bi)=a^2 + b^2
Gauss reinterpreted Euler's theorem on x^2+y^2=p in a remarkable way, but he needed the imaginary number 'i' to do it. He defined new numbers, now called Gaussian integers, as numbers of the form x+iy, where x and y are ordinary integers. For example, 5+3i, 2-i, and 7+0i are Gaussian integers, but 0.35-1.5i is not, because the coefficients in the last one are not ordinary integers, but decimal fractions. Notice that ordinary integers are also a special case of Gaussian integers, with imaginary part 0.
Gauss rewrote Euler's theorem this way: (x+iy)*(x-iy)=p if p is a prime which is 1 (mod 4). That is, if you consider p as a Gaussian integer, then it is no longer prime! It factors, as you see above. Ordinary primes which are 3 (mod 4), on the other hand, are still prime, even when you think of them as Gaussian integers. This reinterpretation of Euler's theorem opened a door to a whole new number system, the Gaussian integers, and hinted at some of its properties. It turns out that there are primes among the Gaussian integers, and every Gaussian integer can be factored uniquely into primes. Gauss's factorization of p when p is 1 (mod 4) turns out to be a factorization into primes, for example.
Here are some examples:
Since 13=9+4 (sum of squares), 13 factors as 13=(3+2i)*(3-2i), and both (3+2i) and (3-2i) are prime in the Gaussian integers.
Since 17=16+1 (sum of squares), 17 factors as 17=(4+i)*(4-i) and both (4+i) and (4-i) are prime in the Gaussian integers.
Entirely analogous statements can be made for any ordinary prime p which is 1 (mod 4).
An amusing thing happens if you have products of such primes, like 221=13*17. Factoring this in the Gaussian integers gives
221=13*17=(3+2i)*(3-2i)*(4+i)*(4-i)=(3+2i)*(4+i)*(3-2i)*(4-i) = (10+11i)*(10-11i)
= 10^2 + 11^2
= (3+2i)*(4-i)*(3-2i)*(4+i) = (14+5i)*(14-5i)
= 14^2 + 5^2
That is, such a product is the sum of two squares in two different ways! The two ways emerge from taking the factorization into Gaussian primes and reordering the factors. A product of 3 primes might be interesting to look at.
Use factorization in the Gaussian integers to represent the following integers as sums of squares in all possible ways:
(1) 65
(2) 69
(3) 85
(4) an integer of your choice.