Problem 74 - Contemporary Precalculus

Question

Describe what happens geometrically when you multiply a complex number by 2.

Step-by-Step Solution

Answer

Short Answer: The geometric effect of multiplying a complex number by 2 is a dilation (stretch) away from the origin by a factor of 2. In the complex plane, the distance between the point representing the complex number and the origin doubles after multiplication.

Step 1: Representation of complex numbers on complex plane

A complex number can be represented as z = a + bi, where a and b are real numbers and i is the imaginary unit (i^2 = -1). In the complex plane, z is represented as a point (a,b). The x-axis corresponds to the real part, a, and the y-axis corresponds to the imaginary part, b.

Step 2: Geometric interpretation of complex number multiplication

When we multiply two complex numbers, say z1 = a1 + b1i and z2 = a2 + b2i, we perform the multiplication as follows: (a1 + b1i)(a2 + b2i) = (a1a2 - b1b2) + (a1b2 + a2b1)i. Geometrically, this can be interpreted as the product of their magnitudes and the sum of their angles (measured from the positive x-axis), also known as their polar representation.

Step 3: Multiplying a complex number by 2

Now, let's consider the multiplication of a complex number z = a + bi by 2. The result is (2a + 2bi), which can be represented by a point (2a, 2b) in the complex plane. Since the real and imaginary parts of the complex number are both multiplied by 2, the distance from the origin of the representation of the complex number 2z doubles compared to the distance of the representation of complex number z. Geometrically, this represents a dilation (stretch) of the complex plane by a factor of 2 along a line connecting the point (a, b) to the origin. In conclusion, when a complex number z is multiplied by 2, its representation as a point in the complex plane is stretched away from the origin by a factor of 2.

Key Concepts

Complex Plane

Imagine a flat, two-dimensional space where every point uniquely represents a complex number. This space is known as the complex plane, a crucial concept in understanding complex numbers. Each complex number corresponds to a point with coordinates (a, b), where 'a' and 'b' are real numbers representing the real and imaginary parts, respectively. The complex plane is akin to a map, where the horizontal axis (the x-axis) marks the real part, while the vertical axis (the y-axis) marks the imaginary part. Just as you might plot a city on a geographic map, you can plot a complex number on this 'map' of complex numbers.

Geometric Interpretation of Complex Multiplication

Complex number multiplication can feel abstract, but it's beautifully straightforward when visualized geometrically. To multiply two complex numbers, we can use their polar coordinates, focusing on the magnitude (distance from the origin) and the angle they make with the positive real axis. The principle behind the geometric interpretation of complex multiplication is that when you multiply two complex numbers, the magnitudes multiply while the angles add. This method not only simplifies calculation but also gives a compelling visual understanding of how complex numbers combine to create new ones.

Polar Representation of Complex Numbers

While the standard form of a complex number is a + bi, the polar representation takes a different route, expressing the number in terms of its magnitude and angle, noted as (r, θ). Here, 'r' is the length of the vector from the origin to the point (a, b) on the complex plane, and 'θ' is the angle that vector makes with the positive real axis.The polar form of a complex number is highly useful, especially in multiplication, as it turns the problem into the simpler task of multiplying lengths and adding angles. The conversion from rectangular coordinates (a, b) to polar coordinates (r, θ) is an essential skill in complex number manipulations, providing a deeper understanding of their properties and behaviors.

Dilation in Complex Plane

In the context of complex numbers, dilation refers to the scaling of a complex number's magnitude without changing its angle on the complex plane. Multiplying a complex number by a real positive scaler, such as 2, exemplifies this transformation. The result is a new complex number whose distance from the origin is twice that of the original, producing an effect of stretching or 'dilation' outward. This process maintains the direction of the original vector (the angle in polar representation stays the same), only altering its length. It's an intuitive concept: doubling the real and imaginary parts of a complex number is like zooming in on a picture, making every detail larger without distorting the image.