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Subtracting complex numbers can sometimes be a bit of a challenge, but with a straightforward approach, it becomes a piece of cake. Think of a complex number as having two parts, a real part, and an imaginary part. To subtract complex numbers, subtract the real part of one from the real part of the other and do the same for the imaginary parts. It's just like subtracting two binomials.

For the given exercise, the subtraction is between a pure imaginary number (15i) and a complex number (12-11i). Remember, subtraction is the addition of a negative. So, we distribute the negative sign across the complex number we are subtracting: 15i - (12 - 11i) = 15i - 12 + 11i. This effectively reverses the sign of each term inside the parentheses, transforming the task into a simpler addition problem.

To tidy up an expression involving complex numbers, we need to combine like terms—a fundamental skill in algebra. Like terms have the same variable raised to the same power. In the realm of complex numbers, like terms mean grouping real parts with real parts and imaginary parts with imaginary parts.

In our exercise, once we've subtracted the complex numbers, we combine the imaginary parts (15i and 11i) together. That gives us 15i + 11i, which simplifies to 26i. The real part in this instance is straightforward—there's just one term (-12), so it remains as is. The combined expression, -12 + 26i, is now cleaner and ready to be presented in standard form.

Imaginary numbers might sound like fanciful math fiction, but they're a vital concept in complex numbers. An imaginary number is defined as a multiple of i, where i is the imaginary unit and is equal to the square root of -1. It's denoted as i because the letter stands for 'imaginary'.

These numbers don't exist on the number line we use for real numbers; instead, they inhabit a different dimension, known as the imaginary axis, when plotted on a complex plane. In the context of our exercise, 15i and 11i are both imaginary numbers, and when we combine them, we are essentially adding along this imaginary axis to find our result.