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Mathematics · Ch 4 — Complex Numbers and Quadratic Equations

Difference of Two Complex Numbers

4.3.2

Difference of Two Complex Numbers

The Meaning of Subtraction

Subtraction of complex numbers is built directly from addition and the concept of the negative of a complex number. For any two complex numbers z1z_1 and z2z_2, the difference z1−z2z_1 - z_2 is defined as:

z1−z2=z1+(−z2)z_1 - z_2 = z_1 + (-z_2)

That is, to subtract z2z_2 from z1z_1, you add the negative of z2z_2 to z1z_1. This mirrors the way subtraction works for real numbers — subtracting a number is the same as adding its opposite.

Note

The negative of a complex number z=a+ibz = a + ib is −z=−a−ib-z = -a - ib. So if z2=a2+ib2z_2 = a_2 + ib_2, then −z2=−a2−ib2-z_2 = -a_2 - ib_2.

How It Works in Practice

Let z1=6+3iz_1 = 6 + 3i and z2=2−iz_2 = 2 - i. Then:

z1−z2=(6+3i)+(−(2−i))=(6+3i)+(−2+i)=(6−2)+(3i+i)=4+4iz_1 - z_2 = (6 + 3i) + (-(2 - i)) = (6 + 3i) + (-2 + i) = (6 - 2) + (3i + i) = 4 + 4i

Now reverse the order: subtract z1z_1 from z2z_2:

z2−z1=(2−i)+(−(6+3i))=(2−i)+(−6−3i)=(2−6)+(−i−3i)=−4−4iz_2 - z_1 = (2 - i) + (-(6 + 3i)) = (2 - i) + (-6 - 3i) = (2 - 6) + (-i - 3i) = -4 - 4i

Notice that z2−z1=−(z1−z2)z_2 - z_1 = -(z_1 - z_2). This is exactly what we expect from subtraction of real numbers — subtraction is not commutative.

The General Formula

If z1=a1+ib1z_1 = a_1 + ib_1 and z2=a2+ib2z_2 = a_2 + ib_2, then:

z1−z2=(a1+ib1)+(−a2−ib2)=(a1−a2)+i(b1−b2)z_1 - z_2 = (a_1 + ib_1) + (-a_2 - ib_2) = (a_1 - a_2) + i(b_1 - b_2)

So the difference of two complex numbers is obtained by subtracting their real parts and their imaginary parts separately.

Watch out

A common mistake is to forget that the minus sign applies to both the real and imaginary parts of z2z_2. For example, (6+3i)−(2−i)(6 + 3i) - (2 - i) is not (6−2)+(3i−i)=4+2i(6 - 2) + (3i - i) = 4 + 2i — that would be incorrect because the imaginary part of 2−i2 - i is −i-i, and subtracting −i-i gives +i+i, not −i-i. Always rewrite as addition of the negative first.

Geometric Interpretation

On the complex plane, subtracting z2z_2 from z1z_1 corresponds to the vector from the point representing z2z_2 to the point representing z1z_1. In other words, if you plot z1z_1 and z2z_2 as points, the complex number z1−z2z_1 - z_2 is the vector that points from z2z_2 to z1z_1. Its magnitude is the distance between the two points.

Key Properties

The textbook does not list separate numbered properties for subtraction alone — it defines subtraction in terms of addition and the negative. However, from this definition, several important facts follow directly: …

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