OEF complex
--- Introduction ---

This module actually contains 28 exercises on complex numbers.

For exercises which ask to reply with a complex number, you must enter the number under the form a+b*i.

Equilateral triangle

If and are points in the complex plane corresponding to and , find the complex number of the point such that the triangle is equilateral.

Asked argument

Give the module and argument of the complex number z=.

Given argument

Compute the real part and imaginary part of the complex number z having |z|=, and having as an argument.

Argument of sum

Let z1=, z2=. Give the module and an argument of z=z1+z2.

CBRT

Let be . What is the number w = +- ?

Cubic root

Find a cubic root w of the complex number z = , by giving its real part and imaginary part.

Equation with module

Solve in C the equation z|z|=.

(Type 0 or 0+0*i if you think that the equation has no solution.)

Fraction

Compute the real part and the imaginary part of the following complex number.

Fraction II

Compute the real part and imaginary part of the complex number .

Maximum of module

What is the of the module |+z|, where z is a complex number with |z|=?

maximal module

Among all complex numbers z with |z|=, find the one for which the module of +z is maximal.

Module of sum

Does there exist two complex numbers and such that ||=, ||=, ||=?

Module of sum II

Let and be two complex numbers with ||=, ||=, and Arg(/)= degrees. Compute the module of  .

Pentaroot

Let ,,, be the 4 (complex) roots of the polynomial X4+X3+X2+X+1. What is the number w=+++ ?

Power conjugate

How many complex numbers z there are, such that z is equal to the conjugate of z?

Pythagorus

Let and be two complex numbers with ||=, ||=, ||=. What is the value of || ?

Pythagorus II

Let and be two complex numbers with ||=, |+|=|-|=. What is the value of || ?

Quadratic double root

For which complex value of the polynomial

X2+()X+

has a (complex) double root?

Quadratic double root II

For which real values of and the polynomial

X2+()X+

has a (complex) double root?

Roots quadratic polynomial

Compute the two roots of the polynomial

P() = 2 + () + ().

You may enter the two roots , in any order.

Root and coefficients

Let P(X)=X2+pX+q be a polynomial with real coefficients p and q. One knows that P has a complex root whose imaginary part is equal to . In this case, there is a relation between the coefficients p and q. Determine this relation by giving q as a function of p.

Square root

Find a square root w of the complex number z = , by giving its real part and imaginary part.

Sum with inverse

Let be a complex number with z+1/z=. What is the number w=+- ?

Sum of i

Compute the sum

S=+++...+.

Sum of j

Sum of unit roots

Two roots

Set , be the two roots of a polynomial P(X)=X2+pX+q, where p is , q is real. Suppose that and are neither real nor pure imaginary. Then the number - is ____________.

Two roots II

Let , be the two roots of a polynomial P(X)=X2+q, where q is a real number. Then - is a ____________ number.

Other exercises on: complex numbers  

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Description: collection of exercises on complex numbers. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games

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