Subtract 3 2i from 9 8i – Embark on a mathematical adventure as we delve into the captivating realm of complex number subtraction. Join us as we unravel the intricacies of subtracting 3 + 2i from 9 + 8i, unveiling a world of numbers that dance beyond the confines of the real.
Prepare to witness the harmonious interplay of real and imaginary parts, the significance of conjugate pairs, and the geometric elegance of complex numbers on the complex plane. Our journey promises to illuminate the intricacies of this fascinating mathematical operation, leaving you with a profound understanding and a newfound appreciation for the beauty of complex numbers.
Complex Number Subtraction
Subtracting complex numbers involves subtracting the real parts and imaginary parts separately. Here’s how you do it:
Subtracting Complex Numbers
- Subtract the real parts: (a
- b) + (c
- d)i
- Subtract the imaginary parts: (a
- b) + (c
- d)i
Example
Let’s subtract 3 + 2i from 9 + 8i:
(9 + 8i)
- (3 + 2i) = (9
- 3) + (8
- 2)i = 6 + 6i
Real and Imaginary Parts
Complex numbers have two components: the real part and the imaginary part. The real part is the part that does not contain the imaginary unit i, while the imaginary part is the part that does contain i. For example, in the complex number 3 + 4i, 3 is the real part and 4i is the imaginary part.
Subtracting Real and Imaginary Parts
When subtracting complex numbers, we subtract the real parts separately from the imaginary parts. For example, to subtract 3 + 2i from 9 + 8i, we first subtract the real parts: 9 – 3 = 6. Then we subtract the imaginary parts: 8 – 2 = 6. So the result of the subtraction is 6 + 6i.
Conjugate Pairs: Subtract 3 2i From 9 8i
In the realm of complex numbers, conjugate pairs hold a special significance, particularly in the context of subtraction. A conjugate pair consists of two complex numbers that are mirror images of each other across the real axis.
The conjugate of a complex number a + bi is defined as a – bi. The real parts are identical, while the imaginary parts have opposite signs. This concept plays a crucial role in complex number subtraction.
Conjugate Pairs in Subtraction
When subtracting complex numbers, the conjugate of the subtrahend (the number being subtracted) can simplify the process. To subtract b – ci from a + di, we can rewrite the subtrahend as a – di (its conjugate) and then add the two complex numbers:
(a + di) + (a
The subtraction of 3 + 2i from 9 + 8i yields a result that can be further explored in Abeka Algebra 2 Quiz 17 . This resource provides detailed explanations and practice problems to enhance your understanding of complex number operations, including subtraction.
By delving into the quiz, you can reinforce your knowledge and master the concept of subtracting complex numbers like 3 + 2i from 9 + 8i.
- di) = (a + a) + (di
- di) = 2a
This method eliminates the imaginary part of the subtrahend, making the subtraction process more straightforward.
Example
Consider the task of subtracting 3 + 2i from 9 + 8i. We can find the conjugate of 3 + 2i as 3 – 2i and then perform the subtraction:
(9 + 8i)
- (3 + 2i) = (9
- 3) + (8i
- 2i) = 6 + 6i
Table of Operations
To make the process of subtracting complex numbers clearer, let’s create a table that Artikels each step involved.
Table of Operations, Subtract 3 2i from 9 8i
| Original Complex Numbers | Real Part Subtraction | Imaginary Part Subtraction | Resulting Complex Number |
|---|---|---|---|
| 9 + 8i | 9
|
8
|
6 + 6i |
As you can see from the table, subtracting complex numbers involves subtracting the real parts and imaginary parts separately.
Geometric Interpretation
Complex numbers can be represented geometrically on the complex plane, where the real part is plotted on the horizontal axis and the imaginary part is plotted on the vertical axis. Each complex number is represented by a point on the plane.
Subtracting complex numbers can be visualized geometrically by translating the second complex number so that its tail is at the head of the first complex number. The resulting vector represents the difference between the two complex numbers.
Subtracting 3 + 2i from 9 + 8i on the Complex Plane
To subtract 3 + 2i from 9 + 8i, we translate the vector 3 + 2i so that its tail is at the head of the vector 9 + 8i. The resulting vector is 6 – 6i, which represents the difference between the two complex numbers.
- The original complex numbers are represented by the points (9, 8) and (3, 2) on the complex plane.
- We translate the vector 3 + 2i so that its tail is at the head of the vector 9 + 8i. This gives us the vector 6 – 6i.
- The vector 6 – 6i represents the difference between the two complex numbers, which is 6 – 6i.
FAQ Guide
What is the significance of conjugate pairs in complex number subtraction?
Conjugate pairs play a crucial role in complex number subtraction as they simplify the process by allowing us to subtract the real and imaginary parts separately.
Can you provide a visual representation of subtracting 3 + 2i from 9 + 8i on the complex plane?
Certainly! On the complex plane, 3 + 2i would be represented as a point 3 units to the right of the origin and 2 units above it. Subtracting 3 + 2i from 9 + 8i would involve moving 3 units to the left and 2 units down, resulting in a final point at 6 + 6i.
