11th Grade - Complex numbers

11th Grade lesson

7 - Complex numbers

Complex numbers are simple.

i is an imaginary number such as property number i.
complex number is a complex number, it has a real part (a) and an imaginary part (b). a and b are real numbers.

Calculations with complex numbers

- complex numbers calculation

- To write the complex number complex number as algebraic form (algebraic form), multiply both numerator and denominator by the conjugate of the denominator. If algebraic form, the conjugate of z is the complex number conjugate complex number. Hence :
complex number calculation

Complex numbers in the plane

complex numbers in the plane

In the complex plane we no longer talk about coordinates but affixes. The point A is no longer located by two coordinates but with only one affix that is a complex number. Here L is a point that has affix complex number affix, I is a point whose affix is complex number affix i, and T is a point whose affix is complex number affix t.
The notion of polar coordinates works well in the complex plane, but there is some new vocabulary.
If M is a point of the plane of affix z, the modulus of z, or the absolute value of z, (denoted by modulus complex number), is the distance OM, and the argument of z (denoted by argument complex number), is the angle argument et angle. If nombres complexs we get :
complex numbers
complex numbers

These formulas enable us to calculate the modulus and the argument of a complex number. When you've calculated the modulus and the argument, you can write the complex number in its polar form :

trigonometric form

Or exponential form :
exponential form

Properties of the modulus and the argument

The modulus of a product is the product of the modulus and the argument of a product is the sum of the arguments : if z and z' are two complex numbers :
property modulus and argument

Distances and angles

If the affix of point A is affix point and the affix of point B is affix point, then vector vector has affix affix vector. It's similar to coordinates.
Now let's place a point M such as affix point c.

Because complex numbers property, the point M has affix affix point. Hence math equation, thus to calculate distances in the complex plane, we have the formula :
math equation

Now let's add on the drawing two points C and D whose affixes respectively are affixes points complexs and math complexs numbers.
We get math equation, hence math equation.


Similarly, complex numbers calculation math hence complex numbers calculation math.
Since complex numbers calculation math , finally :
complex numbers calculation

You can use the formula with any point, and usually, to calculate an angle in the complex plane, we use the formula :

angle and argument

Transformations in the complex plane

There are formulas that enable us to calculate, in the complex plane, the affix of the image of a point by a translation, an homothecy or a rotation. If the affix of M is z, if omega has affix omega, if vector is a vector of affix t, then the image M' of M by the translation of vector vector has affix translation complex, the image of M by the homothecy of center omega and ratio k has affix homothetie complex, and the image of M by the rotation of angle angle alpha and center omega has affix complex rotation. You have to know these formulas very well !

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Complex numbers

lesson, problems