GEC 210 Complex Number

What is complex number

Numbers such as integer,rational number e.t.c that we have been using so far are called real numbers.They can be used to count with or measure distances ,time e.t.c

Such as:22=4, (-2)2=+4√(4)=+/- 2

In other case such as solving x2+1=0 or x2=-1

This has no real solution but by introducing the symbol j=√(-1)

We can write as x =+/- j

The object j is sometimes called imaginary number and is an example of a complex number. There is ,however nothing imaginary about it. While it certainly not represent a quantity in the normal sense that a real number such as 2 does. Having defined it as above it now allows us to write down the solution of any quadratic equation.

Cartesian Complex Number

Example

example

The Argand diagram

A complex number may be represented pictoriallyon rectangular or Cartesian axes. The horizontal (orx) axis is used to represent the real axis and the

vertical (or y) axis is used to represent the imaginary axis. Such a diagram is called an Argand diagram. the point A represents the complex number (3+j2) and is obtained by plotting the coordinates (3, j2) as in graphical work.

The Argand diagram

Addition and subtraction ofcomplex numbers

• Two complex numbers are added/subtracted by adding/subtracting separately the two real parts and the two imaginary parts.

Examples

The Argand diagram

The Argand diagram

• Multiplication of complex numbers is achieved by assuming all quantities involved are real and then using j2 = - 1 to simplify.

Examples

Complex conjugate

• The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. Hence the complex conjugate of a + jb is a - jb. The product of a complex number and its complex conjugate is always a real number.

Example

Division of complex numbers

• Division of complex numbers is achieved by multiplying both numerator and

denominator by the complex conjugate of the denominator.

Example

Examples

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Complex Equations

• If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal. Hence if a + jb = c + jd, then a = c and b = d.

Examples

Polar form of a complex number

• Let a complex number Z be x+jy as shown in the Argand diagram below.

Let distance OZ be r and the angle OZ makes with the positive real axis be Ѳ.

From trigonometry, x = r cos Ѳ and y = r sin Ѳ Hence Z = x + jy = r cos Ѳ + j(r sin Ѳ) = r(cos Ѳ + j sin Ѳ) Z = r(cos Ѳ +j sin Ѳ ) is usually abbreviated to Z = which is known as the polar form of a complex number.

• r is called the modulus (or magnitude) of Z and is written as mod Z or IZI.

r is determined using Pythagoras’ theorem on triangle OAZ

• Determine the modulus and argument of the complex number Z = 2 + j3, and express Z in polar form

Multiplication and division inpolar form

EDe Moivre’s theorem

• From multiplication of complex numbers in polar• form,• (r θ) × (r θ) = r∠ ∠ 2 2θ∠• Similarly, (r θ)×(r θ)×(r θ)=r∠ ∠ ∠ 3 3θ, and so on.∠• In general, De Moivre’s theorem states:• [r∠θ]n =rn n∠ θ• The theorem is true for all positive, negative and

fractional values of n. The theorem is used to determine powers and roots of complex numbers.

Powers of complex numbers

Roots of complex numbers

From the Argand diagram shown below thetwo roots are seen to be 180◦ apart, which is alwaystrue when finding square roots of complex numbers.

• In general, when finding the nth root of a complex number, there are n solutions. For example, there are three solutions to a cube root, five solutions to a fifth root, and so on. In the solutions to the roots of complex number, the modulus, r, is always the same, but the arguments, θ, are different. It is shown in the example that arguments are symmetrically spaced on an Argand diagram and are (360/n)◦ apart, where n is the number of the roots required. Thus if one of the solutions to the cube root of a complex number is, say, 5 20◦, the ∠other two roots are symmetrically spaced (360/3)◦, i.e. 120◦ from this root and the three roots are 5 20◦, 5 140◦ and 5 260◦.∠ ∠ ∠

The exponential form of a complex numberCertain mathematical functions may be expressed as power series.