How to Graph Complex Numbers

How to Graph Complex Numbers

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The Complex Plane

Graph \(a+bi\) as the point \((a,b)\). The horizontal axis is real and the vertical axis is imaginary.

Modulus and Distance

The modulus is \(|a+bi|=\\sqrt{a^2+b^2}\), the distance from the origin.

Coordinate Skills

Midpoints, distances, and translations work the same way they do on the coordinate plane.

Reference Graphs and Visuals

Graphing Complex Number

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Graphing Complex Numbers

Think of this lesson as more than a rule to memorize. Graphing Complex Numbers is about real and imaginary parts working together. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Complex numbers use \(i^2=-1\). Treat \(a+bi\) like a two-part number: the real part \(a\) and the imaginary part \(b\).

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

  • Read the scale and labels first.
  • Identify the key values the graph shows.
  • Connect the graph to the formula or data table.
  • Answer using the units and context.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Graphing Complex Numbers

1) \(Find 4+3i.\)

2) \(Find -2+5i.\)

3) \(Find 3-7i.\)

4) \(Find -6-2i.\)

5) \(Find |3+4i|.\)

6) \(Find |-5+12i|.\)

7) \(Find \text{point }(7,-1).\)

8) \(Find \text{point }(-4,0).\)

9) \(Find \text{distance between }1+2i\text{ and }4+6i.\)

10) \(Find \text{midpoint of }2+8i\text{ and }6-4i.\)

11) \(Find -z\text{ if }z=-3+4i.\)

12) \(Find z+3i\text{ if }z=5-2i.\)

13) \(Find |-8-6i|.\)

14) \(Find \text{numbers 5 units from origin among }5i,3+4i,6-i.\)

15) \(Find \text{distance between }-2+3i\text{ and }4-5i.\)

16) \(Find \text{midpoint of }-1+i\text{ and }7+9i.\)

17) \(Find |a+12i|=13,\ a>0.\)

18) \(Find \text{real-axis points 4 units from }3+0i.\)

19) \(Find z+(4+5i)\text{ if }z=2-3i.\)

20) \(Find |z-(2-i)|=5.\)

 

1)\(Find 4+3i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: real part is x, imaginary coefficient is y.

Answer: \((4,3)\)

2)\(Find -2+5i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: real part is -2 and imaginary coefficient is 5.

Answer: \((-2,5)\)

3)\(Find 3-7i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: the point is (3,-7).

Answer: \(\text{Quadrant IV}\)

4)\(Find -6-2i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: the point is (-6,-2).

Answer: \(\text{Quadrant III}\)

5)\(Find |3+4i|.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: \\sqrt{3^2+4^2}=5.

Answer: \(5\)

6)\(Find |-5+12i|.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: \\sqrt{25+144}=13.

Answer: \(13\)

7)\(Find \text{point }(7,-1).\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: write x+yi.

Answer: \(7-i\)

8)\(Find \text{point }(-4,0).\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: the imaginary part is 0.

Answer: \(-4\)

9)\(Find \text{distance between }1+2i\text{ and }4+6i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: \\sqrt{3^2+4^2}=5.

Answer: \(5\)

10)\(Find \text{midpoint of }2+8i\text{ and }6-4i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: average both coordinates.

Answer: \(4+2i\)

11)\(Find -z\text{ if }z=-3+4i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: multiply both parts by -1.

Answer: \(3-4i\)

12)\(Find z+3i\text{ if }z=5-2i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: add imaginary coefficients.

Answer: \(5+i\)

13)\(Find |-8-6i|.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: \\sqrt{64+36}=10.

Answer: \(10\)

14)\(Find \text{numbers 5 units from origin among }5i,3+4i,6-i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: their moduli are 5, 5, and sqrt(37).

Answer: \(5i\text{ and }3+4i\)

15)\(Find \text{distance between }-2+3i\text{ and }4-5i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: \\sqrt{6^2+(-8)^2}=10.

Answer: \(10\)

16)\(Find \text{midpoint of }-1+i\text{ and }7+9i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: average real parts and imaginary coefficients.

Answer: \(3+5i\)

17)\(Find |a+12i|=13,\ a>0.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: a^2+144=169.

Answer: \(a=5\)

18)\(Find \text{real-axis points 4 units from }3+0i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: |x-3|=4.

Answer: \(-1\text{ and }7\)

19)\(Find z+(4+5i)\text{ if }z=2-3i.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: sum is 6+2i.

Answer: \((6,2)\)

20)\(Find |z-(2-i)|=5.\)

Step 1: Interpret \(a+bi\) as the point \((a,b)\) when graphing.

Step 2: this is a circle in the complex plane.

Answer: \(\text{center }(2,-1),\ \text{radius }5\)

Graphing Complex Numbers Practice Quiz