i is an imaginary number such as .
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-
- To write the complex number as (algebraic form), multiply both numerator and denominator by the conjugate of the denominator. If , the conjugate of z is the complex number . Hence :
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 , I is a point whose affix is , and T is a point whose affix is .
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 ), is the distance OM, and the argument of z (denoted by ), is the angle . If we get :
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 :
Or exponential form :
Properties of the modulus and the argumentThe 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 :
Distances and anglesIf the affix of point A is and the affix of point B is , then vector has affix . It's similar to coordinates.
Now let's place a point M such as .
Because , the point M has affix . Hence , thus to calculate distances in the complex plane, we have the formula :
Now let's add on the drawing two points C and D whose affixes respectively are and .
We get , hence .
Similarly, hence .
Since , finally :
You can use the formula with any point, and usually, to calculate an angle in the complex plane, we use the formula :
Transformations in the complex planeThere 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 has affix , if is a vector of affix t, then the image M' of M by the translation of vector has affix , the image of M by the homothecy of center and ratio k has affix , and the image of M by the rotation of angle and center has affix . You have to know these formulas very well !