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Imaginary numbers are an extension of the real number system. They come into play when we deal with square roots of negative numbers. Normally, we can't find real numbers whose square is negative. That's where imaginary numbers help us.
The basic imaginary unit is denoted by 'i', where by definition, \( i^2 = -1 \). This means \( i \) is the square root of -1.
Whenever we have a negative number under a square root, we take out the \( i \), allowing us to work with the positive value instead. For example, \( \sqrt{-11} \) becomes \( \sqrt{11}i \). This conversion helps us to use imaginary numbers in calculations when negative square roots are involved.

Complex numbers are a combination of real and imaginary numbers. They are written in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part.
In many mathematical and engineering fields, complex numbers are crucial because they provide solutions to equations that don't have real solutions. A real number alone is actually a complex number with the imaginary part equal to zero.
When we talk about complex numbers, think of them as two-dimensional numbers. These dimensions come from the real part and the imaginary part. They can be visualized on a plane, called the complex plane, where the x-axis represents the real part and the y-axis the imaginary part.

Simplifying expressions is an essential skill in math that involves breaking down complex expressions into simpler or more manageable forms. This usually makes calculations easier and solutions more clear.
When simplifying, follow these steps:

• Perform operations inside brackets or parentheses first.
• Simplify any powers or roots.
• Combine like terms, if possible.
• Finally, ensure the expression is fully reduced and no further simplification can be done.

For our exercise, as we simplified the expression \( \sqrt{25-36} \), we first calculated \( 25 - 36 \) to get \( -11 \). Recognizing that \( \sqrt{-11} \) is not a real number, we introduced \( i \) to handle the square root of the negative number, arriving at \( \sqrt{11}i \). Simplification shows us that understanding the parts of a mathematical expression deeply can lead to more accurate and executable results.