Chapter 1 - Complex Numbers 1 Flashcards
i and Argand Diagrams, this chapter is definitely not imaginary...
What is the value of the imaginary number, i?
The value of i is √-1.
What is the complex number form?
Complex numbers are written in the form: a + bi, where, a is the real number component and b is the coefficient of i, the imaginary component.
What is the value of i²?
i² equals -1 (minus one).
What is the complex conjugate?
For a number in the form, a+bi, the complex conjugate is the same number, with the operator switched. (a-bi)
What is true for a binomial equation, if a root is in the form, a+bi?
If a binomial equation has the root, a+bi, then the complex conjugate is also the root.
How do you represent a point on an Argand Diagram, using a number in the form of a+bi?
An Argand diagram has two axis: the x axis is the Real Axis, and the y axis is the Imaginary axis. If the point is represented by the number a+bi, then the coordinates of that point are (a,b).
If an equation has multiple roots, of which there are some that have the form a+bi, how would one represent this on an Argand Diagram?
If one has multiple points, represented by numbers in the form a+bi, these points can be plotted to make a line or a shape.
How do you represent the modulus of a complex number? And what does the modulus represent?
The modulus of a complex number, |z|, in the form a+bi, is given as √(a² + b²). The modulus represents the length of the vector, from the Origin.
How do you represent the argument of a complex number? And what does the argument represent?
The argument of a complex number, in the form a+bi, is the angle of the line relative to the imaginary axis, where theta is between -π and π. (-π < θ ≤ π)
What are the two argument rules for complex numbers?
arg(z₁ x z₂) = arg(z₁) + arg(z₂)
arg(z₁ ÷ z₂) = arg(z₁) - arg(z₂)
What is the modulus argument form of a complex number, in the form a+bi?
The modulus argument form of a complex number z = a+bi is given by z = r(cos(θ) + isin(θ)) where r is the modulus of z and theta is the argument.
What are the two modulus rules for complex numbers?
How do you represent a circle using a complex number in the form a+bi?
A locus of points satisfying |z - (a+bi)| = r will be a circle of centre (a,b), and radius r.
How do you represent a half line locus using a complex number in the form a+bi?
The locus of points satisfying arg(z - (a+bi)) = θ is a half line from point (a,b) at an angle theta to the positive real axis.
How do you represent a perpendicular bisector locus using a complex number in the form a+bi?
The locus of points satisfying |z-(a+bi)| = |z-(c+di)| is the perpendicular bisector of the line joining (a,b) and (c,d)
