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Imaginary Part

Imaginary Part

The coefficient of i in a complex number. For a complex number a + bi, the imaginary part is b.

 

Table showing complex numbers and their imaginary parts: 6−2i→−2, i−3→1, 5i→5, 10+√2→0

 

 

See also

Real part

Key Formula

If z=a+bi, then Im⁡(z)=b\text{If } z = a + bi, \text{ then } \operatorname{Im}(z) = b
Where:
  • zz = A complex number
  • aa = The real part of z (a real number)
  • bb = The imaginary part of z (a real number)
  • ii = The imaginary unit, where i² = −1

Worked Example

Problem: Find the imaginary part of the complex number z = −5 + 12i.
Step 1: Write the complex number in standard form a + bi.
z=−5+12iz = -5 + 12i
Step 2: Identify the coefficient of i. Here a = −5 and b = 12.
a=−5,b=12a = -5, \quad b = 12
Step 3: The imaginary part is the value of b.
Im⁡(z)=12\operatorname{Im}(z) = 12
Answer: The imaginary part of −5 + 12i is 12.

Another Example

Problem: Find the imaginary part of z = 4 − 9i.
Step 1: Rewrite in standard form a + bi by treating the subtraction as addition of a negative.
z=4+(−9)iz = 4 + (-9)i
Step 2: Read off the coefficient of i.
b=−9b = -9
Step 3: State the imaginary part.
Im⁡(z)=−9\operatorname{Im}(z) = -9
Answer: The imaginary part of 4 − 9i is −9 (negative, because the term is −9i).

Frequently Asked Questions

Is the imaginary part b or bi?
The imaginary part is b, not bi. It is the real-valued coefficient that multiplies i. So for 3 + 7i, the imaginary part is the number 7, not the expression 7i. The notation Im(z) always gives a real number.
Can the imaginary part be zero?
Yes. When the imaginary part equals zero, the complex number reduces to a purely real number. For example, 6 + 0i is simply the real number 6, and Im(6) = 0. Every real number is a complex number whose imaginary part is zero.

Real Part vs. Imaginary Part

For a complex number z = a + bi, the real part is a and the imaginary part is b. Both are ordinary real numbers. The real part captures the position along the horizontal axis of the complex plane, while the imaginary part captures the position along the vertical axis. Together, they fully determine the complex number, much like x- and y-coordinates determine a point.

Why It Matters

The imaginary part lets you break a complex number into two real components, which is essential for adding, comparing, and graphing complex numbers. In the complex plane, the imaginary part gives the vertical coordinate, so plotting and geometric reasoning depend on it. Fields like electrical engineering use the imaginary part to represent quantities such as reactance in AC circuits, making it far more than an abstract idea.

Common Mistakes

Mistake: Saying the imaginary part of 3 + 7i is 7i instead of 7.
Correction: The imaginary part is defined as the coefficient b, which is a real number. The term 7i is called the imaginary term, but Im(z) = 7.
Mistake: Forgetting the sign when the imaginary term is subtracted, e.g., writing Im(4 − 9i) = 9.
Correction: Rewrite 4 − 9i as 4 + (−9)i first. The imaginary part is −9, not 9. Always preserve the sign of the coefficient.

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