Glossary
Argument (Angle) (of a complex number)
The angle θ measured counterclockwise from the positive real axis to the line segment connecting the origin to the complex number on the complex plane.
Example:
The Argument (Angle) (of a complex number) for 1 + i is π/4, as it forms a 45-degree angle with the positive real axis.
Cartesian Coordinates
A system for locating points using perpendicular distances from two axes, typically denoted as (x, y).
Example:
The point (3, 4) on a standard graph is expressed using Cartesian Coordinates.
Cartesian to Polar Conversion
The process of transforming Cartesian coordinates (x, y) into polar coordinates (r, θ) using r = √(x² + y²) and θ = tan⁻¹(y/x), with careful quadrant adjustment.
Example:
To express the point (-1, 1) in polar form, you'd use Cartesian to Polar Conversion to find r = √2 and θ = 3π/4.
Complex Numbers
Numbers that extend real numbers by including an imaginary unit 'i', expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part.
Example:
The number 3 + 4i is a Complex Number, combining a real part (3) and an imaginary part (4i).
Complex Plane
A two-dimensional graph where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number.
Example:
The complex number 3 - 4i can be plotted as the point (3, -4) on the Complex Plane.
Imaginary Unit (i)
The fundamental component of complex numbers, defined as the square root of -1 (i = √-1).
Example:
When solving x² + 1 = 0, the solutions are x = ±i (Imaginary Unit).
Imaginary part (of a complex number)
The component 'b' (the coefficient of 'i') in the standard form a + bi, representing the imaginary number portion of a complex number.
Example:
For the complex number -2 + 5i, the Imaginary part (of a complex number) is 5.
Magnitude (Modulus) (of a complex number)
The distance from the origin to the point representing the complex number on the complex plane, calculated as r = √(a² + b²).
Example:
For the complex number 3 + 4i, its Magnitude (Modulus) (of a complex number) is √(3² + 4²) = 5.
Multiple polar coordinate representations
The concept that a single point can be described by more than one set of polar coordinates due to the periodic nature of angles and the ability to use negative radial distances.
Example:
The point (2, π/4) can also be represented as (2, 9π/4) or (-2, 5π/4) due to Multiple polar coordinate representations.
Polar Axis
The positive x-axis in the polar coordinate system, serving as the reference line for measuring angles.
Example:
An angle of 0 degrees or 0 radians is always measured along the positive x-axis, which is the Polar Axis.
Polar Coordinates
A system for locating points using a distance (r) from the origin (pole) and an angle (θ) from the positive x-axis (polar axis).
Example:
To describe a point 5 units away at a 30-degree angle, you'd use (Polar Coordinates) (5, 30°).
Polar Representation of Complex Numbers (r(cos θ + i sin θ))
An alternative way to express a complex number using its magnitude (r) and argument (θ), connecting it to polar coordinates.
Example:
The complex number 2(cos(π/3) + i sin(π/3)) is the Polar Representation of Complex Numbers (r(cos θ + i sin θ)) for 1 + i√3.
Polar to Cartesian Conversion
The process of transforming polar coordinates (r, θ) into Cartesian coordinates (x, y) using the formulas x = r cos(θ) and y = r sin(θ).
Example:
To find the x and y values for (4, π/6), you'd use Polar to Cartesian Conversion to get x = 4cos(π/6) = 2√3 and y = 4sin(π/6) = 2.
Pole
The origin (0,0) in the polar coordinate system, from which all radial distances are measured.
Example:
When plotting a point like (3, π/2), the distance of 3 units is measured from the Pole.
Quadrant Adjustment
The necessary modification of the angle obtained from the arctangent function (tan⁻¹) to ensure it corresponds to the correct quadrant of the original Cartesian point.
Example:
If tan⁻¹(y/x) gives -π/4 for a point in Quadrant II, you must apply Quadrant Adjustment by adding π to get 3π/4.
Real part (of a complex number)
The component 'a' in the standard form a + bi, representing the real number portion of a complex number.
Example:
In the complex number 7 + 3i, the Real part (of a complex number) is 7.
Standard form of a complex number (a + bi)
The conventional way to write a complex number, where 'a' represents the real part and 'b' represents the coefficient of the imaginary unit 'i'.
Example:
The expression 5 - 2i is in the Standard form of a complex number (a + bi), with a=5 and b=-2.
r (radial distance)
The distance from the Pole to a point in the polar coordinate system.
Example:
In the polar coordinate (7, π/4), the value of r (radial distance) is 7, indicating the point is 7 units from the origin.
θ (angle)
The angle measured counterclockwise from the Polar Axis to the line segment connecting the Pole to the point.
Example:
For the polar coordinate (5, 60°), the θ (angle) is 60 degrees, indicating its rotational position from the positive x-axis.