# 2 Complex numbers

## 2.3 Complex arithmetic

Arithmetical operations on complex numbers are carried out as for real numbers, except that we replace *i*^{2} by −1 wherever it occurs.

### Example 1

Let *z*_{1} = 1 + 2*i* and *z*_{2} = 3 − 4*i*. Determine the following complex numbers.

- (a)
*z*_{1}+*z*_{2} - (b)
*z*_{1}−*z*_{2} - (c)
*z*_{1}*z*_{2} - (d)

### Solution

Using the usual rules of arithmetic, with the additional property that *i*^{2} = −1, we obtain the following.

- (a)
*z*_{1}+*z*_{2}= (1 + 2*i*) + (3 − 4*i*) = (1 + 3) + (2 − 4)*i*= 4 − 2*i* - (b) z
_{1}− z_{2}= (1 + 2*i*) − (3 − 4*i*) = (1 − 3) + (2 + 4)*i*= − 2 + 6*i* - (c) z
_{1}z_{2}= (1 + 2*i*)(3 − 4*i*) = 3 + 6*i*− 4*i*− 8*i*^{2}= 3 + 2*i*+ 8 = 11+ 2*i* - (d)

### Exercise 8

Determine the following complex numbers.

- (a) (3 − 5
*i*) + (2 + 4*i*) - (b) (2 − 3
*i*)(−3 + 2*i*) - (c) (5 + 3
*i*)^{2} - (d) (1 +
*i*)(7 + 2*i*)(4 −*i*)

### Solution

- (a) (3 − 5
*i*) + (2 + 4*i*) = 5 −*i* - (b) (2 − 3
*i*)(−3 + 2*i*) = −6 + 9*i*+ 4*i*− 6*i*^{2}= 13*i* - (c)
- (d) We find
- so

Example 1 illustrates how we add, subtract and multiply two given complex numbers. We can apply the same methods to two general
complex numbers *z*_{1} = *x*_{1} + *iy*_{1} and *z*_{2} = *x*_{2} + *iy*_{2}, and obtain the following formal definitions of addition, subtraction and multiplication in .

### Definitions

Let *z*_{1} = *x*_{1} + *iy*_{1} and *z*_{2} = *x*_{2} + *iy*_{2} be any complex numbers. Then the following operations can be applied.

**addition** *z*_{1} + *z*_{2} = (*x*_{1} + *x*_{2}) + *i*(*y*_{1} + *y*_{2})

**subtraction** *z*_{1} − *z*_{2} = (*x*_{1} − *x*_{2}) + *i*(*y*_{1} − *y*_{2})

**multiplication** *z*_{1}*z*_{2} = (*x*_{1}*x*_{2} − *y*_{1}*y*_{2}) + *i*(*x*_{1}*y*_{2} + *y*_{1}*x*_{2})

With these definitions, most of the usual rules of algebra still hold, as do many of the familiar algebraic identities. For example,

and

There is no need to remember or look up these formulas. For calculations, the methods of Example 1 may be used.

An obvious omission from this list of definitions is *division*. We return to division after discussing the *complex conjugate* and *modulus* of a complex number.