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To add complex numbers, you simply need to add their respective real and imaginary parts. A complex number has two components: the real part and the imaginary part. For example, in the expression \((4-i)+(2+5i)\), you will add the real parts, 4 and 2, and the imaginary parts, \-1i and 5i, separately.
Addition is pretty straightforward when you break it into these basic steps:

• Add the real parts: 4 and 2, which gives 6.
• Add the imaginary parts: \-1i and 5i, which gives 4i.

After calculating these sums independently, the solution is a new complex number. So, the sum of \((4-i)+(2+5i)\) becomes \(6 + 4i\). Notice how each part, real and imaginary, is calculated separately and then combined to give the final result. This method is logical and consistent, ensuring you correctly handle complex number addition.

Understanding the real and imaginary parts of a complex number is essential when working with them. A complex number is typically written in the form \(a + bi\), where \(a\) is the real part and \(b\) represents the imaginary part, multiplied by the imaginary unit \(i\).

Consider the complex number \(4-i\):

• The real part is 4.
• The imaginary part is \-1i (or just -1 if you disregard the i).

For the number \(2+5i\):

• The real part is 2.
• The imaginary part is 5i.

When solving problems involving complex numbers, it is crucial to distinctly identify each part. This understanding will not only aid in addition but also in other operations involving complex numbers like subtraction, multiplication, and division.

The standard form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part. This format is convenient and universally recognized for expressing complex numbers.

Let's take the result from our previous problem: \(6 + 4i\). Here:

• 6 is the real part.
• 4i is the imaginary part.

Expressing complex numbers in standard form instantly clarifies which components belong to the real and imaginary dimensions. It facilitates ease in computation and consistency across mathematical problems. Remembering this format will also help you perform operations such as addition, subtraction, and even understanding more complex manipulations like finding conjugates or modulus of complex numbers.