复数的引入，概念清晰，通俗易懂

    在数学发展史上，柯西是第一个把复数分为两个部分的数学家。J.Keisler精心撰写的微积分学教材，在二阶微分方程求解章节中，实时地引入复数及其运算法则，通俗易懂、

大家相信，00后大学生阅读这种英文原文不应该感到困难。

This section begins with a review of the complex numbers. Complex numbers are useful in the solution of second order differential equations. The starting point is the imaginary number i, which is the square root of -1. The complex number system is an extension of the real number system that is formed by adding the number i and keeping the usual rules for sums and products. The set of complex numbers, or complex plane, is the set of all numbers of the form

Z =-X+ iy

where x and y are real numbers. The number x is called the real part of z, and y is called the imaginary part of z. A complex number z can be represented by a point in the plane, with the real part drawn on the horizontal axis and the imaginary part on the vertical axis, as in Figure 14.5.1. The sum of two complex numbers is computed in the same way as the sum of two vectors,

Figure 14.5.1

(a+ ib) + (c + id) =(a+ c)+ i(b +d).

iy

----------------... X + iy

The product of two complex numbers is computed using the basic rule f = - 1 and the rules of algebra: (a+ ib) • (c + id) = ac + ibc + iad + i2bd = (ac - bd) + i(bc + ad).

EXAMPLE 1 Compute the product of 3 + i6 and 7 - i. (3 + i6). (7 - i) = (3 • 7 - 6. ( -1)) + i(6. 7 + 3 • ( -1)) = 27 + i39.

The complex conjugate z of z is formed by changing the sign of the imaginary part of z:

a+ ib =a- ib.

The product of a complex……（省略）

袁萌  7月10日