4.6: An extra
Back in Chapter 1 , Problem 6 we introduced the Euclidean algorithm for integers. The same idea was extended to polynomials with integer coefficients in Problem 126 . In both these settings one starts with a domain (whether the set of integers, or the set of all polynomials with integer coefficients) where there is a notion of divisibility: given two elements m , n in the relevant domain, we say
“ n divides m ” if there exists an element q in the domain such that m = qn.
The next problem invites you to think how one might extend the Euclidean algorithm to a new domain, namely the Gaussian integers ℤ[ i ] − the set of all complex numbers a + bi in which the real and imaginary “coordinates” a and b are integers.
Problem 136 Complex numbers a + bi, where both a and b are integers, are called Gaussian integers. Try to formulate a version of the “division algorithm” for “division with remainder” (where the remainder is always “less than” the divisor in some sense) for pairs of Gaussian integers. Extend this to construct a version of the Euclidean algorithm to find the HCF of two given Gaussian integers.
It is a profoundly erroneous truism … that we should cultivate the habit of thinking what we are doing.
The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them.
Alfred North Whitehead (1861–1947)