lesson
Updated 6 days ago Β· 2 views
You can easily place 3 or β5 on a standard number line, but where does a number like 3+4i live?
Because it has both a real part and an imaginary part, a single line isn't enoughβyou need an entire plane.
The Complex Plane
In 1806, mathematician Jean-Robert Argand solved this puzzle by adding a vertical axis for imaginary numbers, creating the complex plane.
The horizontal axis represents the real axis (extRe), while the vertical axis represents the imaginary axis (extIm).
πA clean coordinate grid showing the Complex Plane. The horizontal axis is labeled 'Real Axis (Re)' with tick marks from -4 to 4. The vertical axis is labeled 'Imaginary Axis (Im)' with tick marks labeled -4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i. Text styling is clean (#1e2945). Subtle grid lines (#e6e6e6). A callout box explains: 'Horizontal = Real Part (a)', 'Vertical = Imaginary Part (b)'.
How do you translate a complex number in standard form a+bi into a coordinate on this grid?
Plotting a+bi
For any complex number z=a+bi, you treat the real coefficient a as the horizontal coordinate and the imaginary coefficient b as the vertical coordinate, plotting the point (a,b).
To plot z=3+2i, move 3 units right on the real axis and 2 units up on the imaginary axis.
πAn animated step-by-step diagram plotting z = 3 + 2i and w = -4 - 3i on the complex plane. An animated blue dot starts at the origin (0,0), moves right to 3 on the Real axis (leaving a dashed blue line), then moves up to 2 on the Imaginary axis, placing a solid dot labeled 'z = 3 + 2i (3, 2)'. Next, an orange dot tracks from origin left to -4, down to -3, labeled 'w = -4 - 3i (-4, -3)'. Clear labels for Real (Re) and Imaginary (Im) axes.
A common mistake is writing the vertical coordinate as 2i instead of 2.
The axis itself supplies the i, so your vertical coordinate is always just the real number b.