How to Find Distance and Midpoint of Complex Numbers?
Treat complex numbers as points on the complex plane. Their distance is the modulus of their difference; their midpoint is their arithmetic mean.
Quick answer: The distance between two complex numbers z₁ = a + bi and z₂ = c + di is |z₁ − z₂| = √((a − c)² + (b − d)²). Use only the real numbers b and d, not bi and di. Example: for 3 + 2i and 8 + 5i, the distance is √(5² + 3²) = √34 ≈ 5.83. The midpoint is (z₁ + z₂)/2.
TL;DR: Here's the secret about complex numbers: a + bi plots as the point (a, b) on a regular-looking plane. Once you see that, distance and midpoint formulas feel exactly like the ones you already use in coordinate geometry. The distance between two complex numbers is the absolute value of their difference, and the midpoint is just their average. Same formulas, same vibes, the imaginary part simply takes the seat usually occupied by y. Spot that connection and the whole topic clicks.
Key takeaways:
- Plot \(a+bi\) as the point \((a,b)\) - real part on the \(x\)-axis, imaginary on the \(y\)-axis.
- Distance: \(|z_1-z_2|=\sqrt{(a_1-a_2)^2+(b_1-b_2)^2}\) - it's the regular distance formula.
- Midpoint: \(\frac{z_1+z_2}{2}=\left(\frac{a_1+a_2}{2}+\frac{b_1+b_2}{2}i\right)\).
- Modulus \(|z|=\sqrt{a^2+b^2}\) is just the distance from \(z\) to the origin.
- The complex plane (Argand diagram) is the geometric home for complex-number problems.
What should we do to calculate the distance and midpoint of complex numbers? Here, you will learn more about the calculation of distance and midpoint of complex numbers.
We all know how to calculate the distance between two spots and the midpoint between two points in a Cartesian plane, but what happens when we’re dealing with something more complicated like a complex plane?
Type the problems from this lesson and check each step.
It turns out that the formulae used to get the distance between two complex numbers and the midpoint between two complex numbers are very similar to the formulae used to determine the distance between two Cartesian points.
Complex numbers are discussed in this part, and we will learn how to use both the midpoint formula and the distance formula for them.
Related Topics
A step-by-step guide to finding the distance and midpoint of complex numbers
The following is the formula for calculating the distance between two points \((a, b)\) on the complex plane for the complex number \(a + bi\) and \((c, d)\) for the complex number \(c + di\):
\(\color{blue}{d=\sqrt{(c-a)^{2}+(d-b){^2}}}\)
- It does not matter which number you subtract from which; the distance comes out the same. If you subtract the real component of the first integer from the second, you must also subtract the first number’s imaginary part from the second.
Using the midpoint rule from coordinate geometry, we can get the midpoint of any two complex numbers \(z=a+bi\) and \(w=c+di\):
\(\color{blue}{M=\frac{a+c}{2}+(\frac{b+d}{2})i}\)
- Thus, the complex number at the middle of the line is given.
Finding Distance and Midpoint of Complex Numbers, Example 1:
Find the distance between the points \(3+2i\) and \(8+5i\).
To find the distance between complex numbers use this formula: \(\color{blue}{d=\sqrt{(c-a)^{2}+(d-b){^2}}}\).
The distance is: \(d=\sqrt{(8-3)^2+(5-2)^2}=\sqrt{(5)^2+(3)^2}\)
Then: \(i^2=-1\)
\(d=\sqrt{(5)^2+(3)^2}=\sqrt{25+9}=\sqrt{34}\approx 5.83\)
Finding Distance and Midpoint of Complex Numbers, Example 2:
Find the midpoint between the points \(6-4i\) and \(2+2i\)
To find the midpoint between complex numbers use this formula: \(\color{blue}{M=\frac{a+c}{2}+(\frac{b+d}{2})i}\)
\(M= \frac {6+2}{2}+(\frac {-4+2}{2})i\)\(=\)\( \frac {8}{2}+(\frac {-2}{2})i=4-i\)
Exercises for Finding Distance and Midpoint of Complex Numbers
Find the distance and midpoint of complex numbers.
- \(\color{blue}{(1+2i)}\), \(\color{blue}{(-2+4i)}\)
- \(\color{blue}{(-6+2i)}\), \(\color{blue}{(-2+5i)}\)
- \(\color{blue}{(-7i)}\), \(\color{blue}{(5-4i)}\)
- \(\color{blue}{(2+i)}\), \(\color{blue}{(6+5i)}\)
- \(\color{blue}{(-7-3i)}\), \(\color{blue}{(3+5i)}\)
- \(\color{blue}{d=\sqrt{13}}\), \(\color{blue}{M=-\frac{1}{2}+3i}\)
- \(\color{blue}{d=5}\), \(\color{blue}{M=-4+\frac{7}{2}i}\)
- \(\color{blue}{d=\sqrt{34}}\), \(\color{blue}{M=\frac{5}{2}-\frac{11}{2}i}\)
- \(\color{blue}{d=4\sqrt{2}}\), \(\color{blue}{M=4+3i}\)
- \(\color{blue}{d=2\sqrt{41}}\), \(\color{blue}{M=-2+i}\)
Recommended EffortlessMath Books
For a workbook that builds complex numbers from the ground up, the Pre-Calculus for Beginners walks through complex arithmetic, the complex plane, and polar form with worked examples. For deeper algebra-side coverage of complex numbers, the Algebra II for Beginners covers the same topics with extra practice.
Common Mistake: Do Not Square the i
The distance formula works on points in the complex plane. The number 8 + 5i is the point (8, 5), so its vertical coordinate is 5, not 5i. When you subtract the imaginary parts, use the plain numbers.
- Right: for 3 + 2i and 8 + 5i, the differences are 8 − 3 = 5 and 5 − 2 = 3. Distance = √(25 + 9) = √34.
- Wrong: writing the second difference as 3i and squaring it to get −9. That gives √(25 − 9) = 4, which is too short.
A quick check: the distance can never be smaller than either difference on its own. Here √34 ≈ 5.83 is bigger than both 5 and 3, so it makes sense. Another way to say this is that the distance is the modulus of the difference: (8 + 5i) − (3 + 2i) = 5 + 3i, and |5 + 3i| = √34.
Finding a Missing Endpoint from the Midpoint
Sometimes you know the midpoint and one endpoint. Since M = (z₁ + z₂)/2, you can solve for the other endpoint: z₂ = 2M − z₁.
Example: The midpoint of z₁ and z₂ is 1 + 4i, and z₁ = 3 + 2i. Find z₂.
- Double the midpoint: 2(1 + 4i) = 2 + 8i.
- Subtract z₁: (2 + 8i) − (3 + 2i) = −1 + 6i.
- Check: ((3 + (−1))/2) + ((2 + 6)/2)i = 1 + 4i. It matches.
So z₂ = −1 + 6i.
Frequently Asked Questions
How do you find the distance between two complex numbers?
Subtract the two complex numbers, then find the modulus of the result. For 1 + 2i and 4 + 6i, the difference is 3 + 4i, and its modulus is √(3² + 4²) = √25 = 5. The order of subtraction does not matter, because squaring removes any negative signs. The answer is always a non-negative real number.
What is the distance formula for complex numbers?
For z₁ = a + bi and z₂ = c + di, the distance is |z₁ − z₂| = √((a − c)² + (b − d)²). It is the regular distance formula applied to the points (a, b) and (c, d) on the complex plane. Use only the real coefficients b and d, not bi and di, or the i² will turn into −1 and give a wrong answer.
How do you find the midpoint of complex numbers?
Add the two complex numbers and divide by 2. Combine the real parts and the imaginary parts separately. For 2 + 7i and 6 − 3i, the sum is 8 + 4i, so the midpoint is (8 + 4i)/2 = 4 + 2i. On the complex plane, that is the point (4, 2), halfway between (2, 7) and (6, −3).
What is a modulus in complex numbers?
The modulus of z = a + bi, written |z|, is its distance from the origin on the complex plane: |z| = √(a² + b²). For 5 − 12i, the modulus is √(25 + 144) = √169 = 13. The modulus is always zero or positive, and the distance between two complex numbers is simply the modulus of their difference.
What is the value of i in complex numbers?
The imaginary unit i is defined as √(−1), so i² = −1. Every complex number has the form a + bi, where a is the real part and b is the imaginary part. When you plot a + bi as the point (a, b), the number b is just a coordinate, which is why i never appears inside the distance or midpoint formulas.
Related EffortlessMath Lessons
If a topic on this page feels rusty, these short lessons go deeper:
- How to add and subtract complex numbers (resource currently unavailable)
- How to multiply and divide complex numbers (resource currently unavailable)
- How to use the Pythagorean theorem
- How to use the distance formula
- Axis of symmetry of quadratic functions
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