91Ó°ÊÓ

Complex numbers have a real part and an imaginary part. The standard form of a complex number is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. When adding or subtracting complex numbers, it is important to combine like terms:

• Combine real parts with real parts.
• Combine imaginary parts with imaginary parts.

For example, consider the provided exercise. The initial problem is given as:
\((-2 + 6i) - (-7 + i)\)
First, distribute the negative sign inside the parentheses:
\((-2 + 6i) + 7 - i\)
This transforms our expression into something simpler to combine. Next, combine like terms:

\((-2 + 7) + (6i - i)\)
When combining the real numbers, we get \(5\). When combining the imaginary numbers, we get \(5i\). So, the final answer is:
\(5 + 5i\)

The imaginary unit is a concept in mathematics used to express numbers that cannot be represented on the traditional number line. The imaginary unit, denoted by \(i\), is defined as changing its nature when squared:

\(i^2 = -1\)
You will often encounter imaginary units when dealing with complex numbers.

When performing operations on complex numbers, treat \( i \) just like a variable, but keep in mind its special property \(i^2 = -1\). This property will notably come into play when simplifying expressions involving powers of \(i\).

In our exercise, the imaginary unit plays a key role in the given complex numbers \(6i\) and \(-i\). Notice how these terms operate based on the properties of \(i\): when combined, their coefficients can simply be added or subtracted.

Simplifying expressions involves combining all like terms and ensuring the expression is in its simplest form. In the context of complex numbers, it means making sure the final answer is presented in the form \(a + bi\), where \(a\) and \(b\) represent real numbers.

For the provided problem:
\((-2 + 6i) - (-7 + i)\)

After distributing the negative sign, we get:
\((-2 + 6i) + 7 - i\)

Next, we combine like terms:
\((-2 + 7) + (6i - i)\)
This results in:
\((5) + (5i)\)

So, the simplified expression is \(5 + 5i\).

Always check your final answer to ensure it's in the standard form \(a + bi\). This clear separation of real and imaginary components helps in further operations and understanding of the complex number.