Can you solve the following problems and explain how you obtained the solution?

1. a) (8-2i) + (1-3i) - (15+7i) - (23-6i)

b) (1+i)(4-i)(-3+7i) + (3-2i)(i³-4)

c) ; d)

2. Find Re

3. Find all the complex numbers z such that Re z +Im z = |z|

4. Prove the identity | z₁+z₂|²+|z₁-z₂|²=2(|z₁|²+|z₂|²)

5. Find all the complex numbers z such that and

*Note:

Translate the following example with the explanation into Russian in the written form:

We define a number i by the equation

i²= -1 (1)

The addition of two complex numbers α = a+bi and β = c+di is defined by

α +β =(a+bi)+(c+di)=a+c+(b+d)i (2)

and their multiplication by

α β =(a+bi)(c+di)=ac-bd+(ad+bc)I (3)

The reader will observe that the last result can be obtained by multiplying out the two parentheses and using (1).

Another fact which is easily confirmed from (2) and (3) is that α +β =β +α, i.e., both addition and multiplication are independent of the order in which they are carried out. It is equally obvious that, in the case of three complex numbers α, β, γ, α +(β +γ) is identical with ( α +β)+γ. The parentheses are therefore not necessary and we write the result as α +β +γ. A short computation will confirm that, similarly, α (β γ) = (α β)γ. This product may therefore be written α β γ.

The operations of addition and multiplication are thus both commutative and associative. Since it is easily confirmed that α (β +γ)= α β + α γ, multiplication is also distributive with respect to addition, or, to state it in less elegant language, expressions in parentheses may be handled according to the usual rules.

Give the English equivalents for:

легко подтвердить, не зависят от порядка; аналогично; так как; выражения в скобках; по обычным правилам.

Grammar Notes: