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Complex Numbers: four rules

December 3, 2010 Leave a comment

Consider complex numbers in the form a+bi

Adding complex numbers

When adding complex numbers we add the real part and then we add the imaginary part

Examples

(2+i)+(4+i)=(2+4)+(1+1)i=6+2i

(3+5i)+(-2-3i)=(3-2)+(5-3)i=1+2i

(4-i)+(-4+7i)=(4-4)+(-1+7)i=6i

Subtracting complex numbers

When subtracting complex numbers we subtract the real part and then we subtract the imaginary part

Examples

(2+i)-(4+i)=(2-4)+(1-1)i=-2

(3+5i)-(-2-3i)=(3+2)+(5+3)i=5+8i

(4-i)-(-4+7i)=(4+4)+(-1-7)i=8-8i

Multiplying complex numbers

 When multiplying complex numbers we need to remember that i\times i = -1

Examples

(2+i)(4+i)=8+2i+4i+i^2=7+6i

(3+5i)(-2-3i)=-6-9i-10i-15i^2=9-19i

(4-i)(-4+7i)=-16+28i+4i-7i^2=-9+32i

Dividing complex numbers

Dividing complex numbers requires us to get rid of the imaginary part from the denominator.

We can do this by multiplying the denominator and numerator by the complex conjugate of the denominator.

Examples

\displaystyle z=\frac{2+8i}{1+i}

\displaystyle \implies z=\frac{2+8i}{1+i}\times \frac{1-i}{1-i}

\displaystyle \implies z=\frac{10+6i}{2}=5+3i


\displaystyle w=\frac{3-i}{4+i}

\displaystyle \implies w=\frac{3-i}{4+i}\times \frac{4-i}{4-i}

\displaystyle \implies w=\frac{11-7i}{17}

\displaystyle \implies w=\frac{11}{17}- \frac{7}{17}i

Categories: Complex numbers

Sketching derivative graphs (cubics)

December 1, 2010 Leave a comment

The important thing to remember here is that we are doing a sketch.  It does not have to be accurate, but id does have to conatian all the key points.

Hints:

  1. A cubic differentiates to a quadratic (parabola)
  2. A the turning points have a gradient of zero and will move to the x axis on the derivative sketch
  3. A point with a positive gradient will be above the x axis in the sketch
  4. A point with a pnegative gradient will be below the x axis in the sketch

Examples

Describe and sketch the derivative graph for the function below

The critical points are x=0\mbox{ and }x=\frac{4}{3}, because this is where the turning points occur.

These points will be on the x axis in the sketch.

For x<0 the gradient is positive.

These points will be above the x axis in the sketch.

For 0<x<\frac{4}{3} the gradient is negative .

These points will be below the x axis in the sketch.

The graph is a cubic so the sketch will be a parabola

Practice

Categories: Differentiation