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Imaginary numbers arise when we need to take the square root of a negative number. In the real number system, this isn't possible, so mathematicians introduced a new number, which they call 'i'.
'i' is defined as \(i = \sqrt{-1}\).
The key property of 'i' is that \(i^2 = -1\).
This allows us to express the square root of any negative number as some multiple of 'i'. For example, \(\sqrt{-4} = 2i\) because \(\sqrt{-4} = \sqrt{4 \cdot (-1)} = \sqrt{4} \cdot \sqrt{-1} = 2i\).
In practice, whenever you see a negative number under a square root, you can convert it into an imaginary number using these properties.

Complex numbers extend the idea of imaginary numbers to include both real and imaginary parts. A complex number has the form \(a + bi\), where 'a' and 'b' are real numbers, and 'i' is the imaginary unit.
In the quadratic formula exercise, the final solution was \(0.4 + 0.2i\).
Here, '0.4' is the real part and '0.2i' is the imaginary part.
Complex numbers can be added, subtracted, and multiplied. For instance:

• \((a + bi) + (c + di) = (a+c) + (b+d)i\)
• \((a + bi) - (c + di) = (a-c) + (b-d)i\)
• \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\)

Understanding complex numbers is essential as they frequently arise in engineering, physics, and advanced mathematics.

Simplification involves rewriting an expression in a simpler or more aesthetically pleasing form. Considering our quadratic expression, let's break down the simplification steps:
1. **Simplify the inner expression**: Before taking the square root, simplify any arithmetic inside the square root. For the given exercise, you found \((-4)^{2} - 4(5)(1) = 16 - 20 = -4\).
2. **Evaluate the square root**: Once simplified, evaluate the square root. For \sqrt{-4}\, this becomes `2i`. Knowing the properties of imaginary numbers helps here.
3. **Simplify the overall expression**: Combine all parts back into the quadratic formula. \frac{-(-4) + 2i}{2(5)}\ simplifies to \frac{4 + 2i}{10}\.
4. **Divide each part separately**: Finally, divide the real and imaginary parts by the denominator separately. So, \frac{4}{10} + \frac{2i}{10} = 0.4 + 0.2i\.
Simplification is crucial to make complex expressions easier to understand and work with.