Have you ever tried to solve the equation \(x^2 + 1 = 0\)? If you work only with real numbers, this looks impossible. Any real number squared is at least \(0\), so how could it ever be \(-1\)? Yet mathematicians, scientists, and engineers solve equations like this all the time. The secret is to expand our number system beyond the real numbers to something bigger and more powerful: complex numbers.
Complex numbers let us handle problems involving square roots of negative numbers, describe electrical signals, and even rotate shapes in a clean algebraic way. They might sound mysterious, but they are built from ideas you already know: real numbers and algebraic rules. The key new idea is an unusual number called \(i\), defined so that \(i^2 = -1\).
On the real number line, there is no real number \(x\) such that \(x^2 = -1\). That is because for any real \(x\), \(x^2\) is always greater than or equal to \(0\). To solve equations like \(x^2 + 1 = 0\), we introduce a new number.
We define a special number \(i\) (read as “eye”) such that:
\(i^2 = -1\)
This is not a number you can place on the usual real number line between \(-1\) and \(1\). It is something new, but we decide that it follows all the usual algebra rules: it can be added, subtracted, multiplied, and used in powers, just like a variable, except we know exactly that \(i^2 = -1\).
From the definition, we can quickly find some basic powers of \(i\):
If we keep going, we see a repeating pattern:
This cycle is very useful for simplifying expressions involving higher powers of \(i\).
Once we accept \(i\), we can build a new kind of number by combining real numbers and multiples of \(i\). A complex number is any number of the form:
\(a + bi\)
where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with \(i^2 = -1\).
We use these terms:
Some special cases:
We often label a complex number as \(z = a + bi\). Two complex numbers are equal only if their real parts are equal and their imaginary parts are equal. For example, \(2 + 3i = 2 + 3i\), but \(2 + 3i \neq 3 + 2i\) because both the real and imaginary parts must match.
A powerful way to understand complex numbers is to think of them as points in a plane, similar to coordinates in geometry. The complex number \(a + bi\) corresponds to the point \((a, b)\). As shown in [Figure 1], we use a horizontal axis for real parts and a vertical axis for imaginary parts.
We call this the complex plane (or Argand plane):
For example, the complex number \(3 + 2i\) is represented by the point with coordinates \((3, 2)\): move 3 units to the right along the real axis, then 2 units up along the imaginary axis.
Thinking of complex numbers as points or arrows from the origin makes it much easier to understand operations like addition and even multiplication later. When we multiply by \(i\), for example, we are essentially rotating numbers on this plane, a fact that becomes very important in physics and engineering, and it connects back to the idea of plotting them as in [Figure 1].
Adding and subtracting complex numbers is straightforward: you just combine the real parts and the imaginary parts separately. Geometrically, this is like adding vectors in the plane, as the arrows on the complex plane in [Figure 2] show.
Suppose we have two complex numbers:
\[z_1 = a + bi, \quad z_2 = c + di\]
Then:
We simply treat the real and imaginary parts like separate coordinates.
In the complex plane, if you think of \(a + bi\) and \(c + di\) as arrows from the origin, then \(z_1 + z_2\) is found by placing the tail of one arrow at the head of the other and drawing the diagonal of the parallelogram, just like adding displacement vectors. This geometric view in [Figure 2] matches the coordinate-wise addition rule.
Example 1: Adding complex numbers
Add \( (3 + 4i) + (5 - 2i) \).
Step 1: Add the real parts: \(3 + 5 = 8\).
Step 2: Add the imaginary parts: \(4i + (-2i) = 2i\).
Step 3: Combine:
\[(3 + 4i) + (5 - 2i) = 8 + 2i\]
Example 2: Subtracting complex numbers
Subtract \( (1 - 6i) - (4 + 3i) \).
Step 1: Subtract the real parts: \(1 - 4 = -3\).
Step 2: Subtract the imaginary parts: \(-6i - 3i = -9i\).
Step 3: Combine:
\[(1 - 6i) - (4 + 3i) = -3 - 9i\]
Multiplication is where the special rule \(i^2 = -1\) really matters. To multiply two complex numbers, we use the distributive property (FOIL) just as you would with binomials, and then simplify \(i^2\) whenever it appears.
If \(z_1 = a + bi\) and \(z_2 = c + di\), then:
\[z_1 z_2 = (a + bi)(c + di)\]
Expanding gives:
\[(a + bi)(c + di) = ac + adi + bci + bdi^2\]
Now combine like terms and remember that \(i^2 = -1\):
\[ac + adi + bci + bdi^2 = ac + (ad + bc)i + bd(-1)\]
So the product is:
\[z_1 z_2 = (ac - bd) + (ad + bc)i\]
Example 3: Multiplying complex numbers (basic)
Multiply \((2 + 3i)(4 + i)\).
Step 1: Expand using FOIL:
\[(2 + 3i)(4 + i) = 2 \cdot 4 + 2 \cdot i + 3i \cdot 4 + 3i \cdot i\]
Step 2: Simplify each term:
Step 3: Combine real and imaginary parts:
Real parts: \(8 + (-3) = 5\)
Imaginary parts: \(2i + 12i = 14i\)
Final answer:
\[(2 + 3i)(4 + i) = 5 + 14i\]
Example 4: Multiplying complex numbers with negatives
Multiply \((1 - 2i)(3 - 4i)\).
Step 1: Expand:
\[(1 - 2i)(3 - 4i) = 1 \cdot 3 + 1 \cdot (-4i) + (-2i) \cdot 3 + (-2i)(-4i)\]
Step 2: Simplify each term:
Step 3: Combine:
Real parts: \(3 + (-8) = -5\)
Imaginary parts: \(-4i - 6i = -10i\)
So:
\[(1 - 2i)(3 - 4i) = -5 - 10i\]
For a complex number \(z = a + bi\), the complex conjugate is defined as:
\[\overline{z} = a - bi\]
The conjugate has the same real part but the opposite imaginary part. A useful property is that when you multiply a complex number by its conjugate, you get a real number:
\[(a + bi)(a - bi) = a^2 - abi + abi - b^2 i^2 = a^2 - b^2(-1) = a^2 + b^2\]
Notice the imaginary parts cancel out, and \(i^2 = -1\) turns \(-b^2 i^2\) into \(+b^2\). This is important when you later learn to divide complex numbers, because multiplying by the conjugate helps remove \(i\) from a denominator.
Example 5: Product with the conjugate
Let \(z = 3 + 4i\). Then \(\overline{z} = 3 - 4i\). Compute \(z \overline{z}\).
\[(3 + 4i)(3 - 4i) = 3^2 + 4^2 = 9 + 16 = 25\]
The product is the real number \(25\). This pattern \(a^2 + b^2\) appears a lot when dealing with lengths in the complex plane, and it also connects to magnitude in applications discussed later.
Because \(i\) repeats every 4th power, we can simplify any power \(i^n\) by seeing what the remainder is when \(n\) is divided by 4.
The cycle is:
Then it repeats: \(i^5 = i\), \(i^6 = -1\), and so on. To simplify \(i^n\):
Example 6: Simplifying a higher power of \(i\)
Simplify \(i^{23}\).
Step 1: Divide \(23\) by \(4\). We have:
\[23 = 4 \cdot 5 + 3\]
So the remainder is \(3\).
Step 2: Use the cycle. Since the remainder is \(3\),
\[i^{23} = i^3 = -i\]
Example 7: Simplifying an expression with powers of \(i\)
Simplify \(5i^7 - 2i^{10}\).
First, simplify each power:
Now substitute:
\[5i^7 - 2i^{10} = 5(-i) - 2(-1) = -5i + 2\]
We usually write the real part first:
\[5i^7 - 2i^{10} = 2 - 5i\]
Complex numbers might feel abstract, but they show up constantly in modern science and technology. Many of these uses rely on representing quantities as \(a + bi\) and performing the arithmetic operations you have learned.
1. Electrical engineering and AC circuits
Alternating current (AC) in homes and devices changes direction and size over time. Engineers represent voltage and current as complex numbers so that phase shifts (time delays between signals) can be handled using multiplication by \(i\) and rotations in the complex plane, similar to what you visualize in [Figure 1]. Adding two signals corresponds to adding complex numbers, and combining components in circuits uses complex multiplication and conjugates.
2. Waves and signals
Sound waves, radio waves, and Wi‑Fi signals can all be represented as combinations of sine and cosine functions. Using complex numbers, these are neatly combined into expressions involving \(e^{i\theta}\), which behaves like a rotating vector in the complex plane. When engineers mix signals, filter noise, or compress audio, they are often adding, subtracting, and multiplying complex numbers that encode the strength and phase of each frequency.
3. Rotations in the plane
Multiplying a complex number by \(i\) corresponds to a rotation by \(90^\circ\) in the complex plane. More generally, multiplying by certain complex numbers rotates and scales vectors, a concept used in computer graphics, robotics, and control systems. The idea that \(a + bi\) is a point \((a, b)\), as in [Figure 1], helps make sense of these rotations: multiplication is not just “number crunching”; it is transforming positions and movements.
4. Oscillations and vibrations
Whenever something vibrates or oscillates—like a guitar string, a bridge in the wind, or an earthquake wave—complex numbers provide a compact way to describe both the size and timing of these motions. Adding oscillations becomes addition of complex numbers; combining effects can be expressed through multiplication, and conjugates help extract useful real quantities like energies or amplitudes.
• Existence of \(i\): There is a special number \(i\) defined by \(i^2 = -1\). It is not a real number, but it obeys the usual rules of algebra.
• Form of complex numbers: Every complex number can be written as \(a + bi\), where \(a\) and \(b\) are real numbers. The set of all such numbers extends the real number line into a full plane.
• Complex plane: The complex number \(a + bi\) corresponds to the point \((a, b)\) on the complex plane, with \(a\) on the real axis and \(b\) on the imaginary axis. This view connects algebra with geometry and vectors.
• Arithmetic operations: You add and subtract complex numbers by combining real parts and imaginary parts separately. You multiply them using distribution (FOIL) and the rule \(i^2 = -1\), then simplify to the form \(a + bi\).
• Conjugates: The conjugate of \(a + bi\) is \(a - bi\). Their product is always real: \((a + bi)(a - bi) = a^2 + b^2\), a pattern that appears in many applications.
• Powers of \(i\): Powers of \(i\) repeat in a cycle of 4: \(i, -1, -i, 1\). Any power \(i^n\) can be simplified using this cycle.
• Applications: Complex numbers are not “imaginary” in the sense of useless; they are essential tools in engineering, physics, signal processing, and many technologies you rely on every day.
