When we explore complex numbers, it's really cool to see how we can show their operations using pictures.
Think of a complex number as a point on a flat surface called the complex plane.
On this plane:
For example, the complex number ( z = 3 + 4i ) is represented as the point (3, 4) on this plane.
When we add two complex numbers, it’s like adding arrows (vectors).
Let’s say we have two complex numbers:
To add them, we can do:
[ z_1 + z_2 = (2 + 1) + (3 + 2)i = 3 + 5i. ]
On the complex plane, we can plot the points (2, 3) and (1, 2).
To find the new point (3, 5), we start from the origin (0, 0).
We go to (2, 3) first, and then from (2, 3), we move to (3, 5).
This makes a triangle where the two lines from the origin connect the points.
Multiplying complex numbers is a bit different.
Let’s take the same complex numbers as before:
When we multiply them, we get:
[ z_1 z_2 = (1 + i)(1 + i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i. ]
This multiplication means we are rotating and changing the size of the number.
Each complex number has a size or length called magnitude.
We can find out the magnitude using this formula:
[ |z| = \sqrt{x^2 + y^2}. ]
For ( z_1 ), the magnitude is:
[ |z_1| = \sqrt{1^2 + 1^2} = \sqrt{2}. ]
When we multiply complex numbers, we multiply their sizes together and add their angles. This means the result can rotate the point and either stretch it or shrink it.
To wrap it up, looking at complex numbers through a geometric lens helps us understand them better.
This connection between math and shapes makes studying complex numbers really exciting!
When we explore complex numbers, it's really cool to see how we can show their operations using pictures.
Think of a complex number as a point on a flat surface called the complex plane.
On this plane:
For example, the complex number ( z = 3 + 4i ) is represented as the point (3, 4) on this plane.
When we add two complex numbers, it’s like adding arrows (vectors).
Let’s say we have two complex numbers:
To add them, we can do:
[ z_1 + z_2 = (2 + 1) + (3 + 2)i = 3 + 5i. ]
On the complex plane, we can plot the points (2, 3) and (1, 2).
To find the new point (3, 5), we start from the origin (0, 0).
We go to (2, 3) first, and then from (2, 3), we move to (3, 5).
This makes a triangle where the two lines from the origin connect the points.
Multiplying complex numbers is a bit different.
Let’s take the same complex numbers as before:
When we multiply them, we get:
[ z_1 z_2 = (1 + i)(1 + i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i. ]
This multiplication means we are rotating and changing the size of the number.
Each complex number has a size or length called magnitude.
We can find out the magnitude using this formula:
[ |z| = \sqrt{x^2 + y^2}. ]
For ( z_1 ), the magnitude is:
[ |z_1| = \sqrt{1^2 + 1^2} = \sqrt{2}. ]
When we multiply complex numbers, we multiply their sizes together and add their angles. This means the result can rotate the point and either stretch it or shrink it.
To wrap it up, looking at complex numbers through a geometric lens helps us understand them better.
This connection between math and shapes makes studying complex numbers really exciting!