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The concept of an imaginary unit is fundamental in mathematics, especially when dealing with complex numbers. The imaginary unit, denoted as i, is defined as the square root of -1. It is an abstract concept, since no real number squared will give a negative result. Using the imaginary unit, we can write and solve equations that would otherwise be impossible with just real numbers.

For example, the exercise given includes the expression \(\sqrt{-1}\), which is simply \(i\) because by definition \(i^2 = -1\). Once we comprehend this rule, we can confidently engage with complex numbers, knowing that anytime we encounter the square root of a negative number, we have a multiple of \(i\) on our hands.

The real part of a complex number refers to the number that is not multiplied by the imaginary unit \(i\). In standard form, a complex number is written as \(a + bi\), where \(a\) represents the real part. The real part is located on the horizontal axis of the complex plane, which is a two-dimensional plot that helps to visualize complex numbers.

In the textbook exercise solution, the real part of the complex number is the numeral \(2\). This indicates that our complex number is 2 units away from the origin along the horizontal or the 'real' axis in the complex plane.

In contrast to the real part, the imaginary part of a complex number includes the imaginary unit, \(i\). When we write \(a + bi\), the \(bi\) part is the imaginary component, where \(b\) is a real number that quantifies how many units of \(i\) we have. Unlike the real part, which can be visualized on the horizontal axis, the imaginary part corresponds with the vertical axis in the complex plane.

Following our exercise example, the expression \(\sqrt{-1}\) simplifies to \(i\), making \(1i\) or just \(i\) the imaginary part of our complex number. The number before the \(i\) tells us the 'amount' of \(i\) in the complex number, which is one in this case. Therefore, the complex number \(2+i\) consists of a real part, \(2\), and an imaginary part, \(i\).