Question

\((12+3 i)-(7-8 i)\)

Short answer

The result of \((12+3 i)-(7-8 i)\) is \(5 + 11i\).

Step-by-step solution

Identify the real and imaginary parts

In the complex number \(12 + 3i\), the real part is 12 and the imaginary part is \(3i\). In the complex number \(7 - 8i\), the real part is 7 and the imaginary part is \(-8i\).

Subtract the real parts

Subtract the real part of the second number from the real part of the first number: \(12 - 7 = 5\).

Subtract the imaginary parts

Subtract the imaginary part of the second number from the imaginary part of the first number: \(3i - (-8i) = 3i + 8i = 11i\).

Form the result

The result from the subtraction of the real parts becomes the new real part and the result from the subtraction of the imaginary parts becomes the new imaginary part. The final result is \(5 + 11i\).

Key concepts

Understanding the Real Part of Complex Numbers

A complex number is typically expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The real part, \(a\), is simply a regular number, often referred to as the "non-imaginary" portion of the complex number.
This component of a complex number does not involve the imaginary unit \(i\), which is defined as the square root of -1. For instance, in the complex number \(12 + 3i\), the real part is 12.

• The real part can be visualized on the horizontal axis of a complex plane.
• It's useful to distinguish it from the imaginary part because each contributes to different dimensions.

In subtraction tasks involving complex numbers, the real parts are subtracted separately from the imaginary parts.

Identifying the Imaginary Part in Complex Numbers

The imaginary part of a complex number is the term involving the imaginary unit, \(i\). For a complex number represented as \(a + bi\), the imaginary part is \(bi\).
Imaginary numbers play a crucial role in complex numbers, allowing us to solve equations that do not have real solutions.

• The imaginary number \(i\) is defined such that \(i^2 = -1\).
• For the complex number \(7 - 8i\), the imaginary part is \(-8i\).

In operations involving complex numbers, the imaginary parts are handled separately from the real parts, including when performing subtraction. This involves paying careful attention to the signs because subtracting a negative means adding the absolute value, as seen in the calculation \(3i - (-8i) = 3i + 8i = 11i\).

Subtraction of Complex Numbers

Subtracting complex numbers involves dealing separately with their real and imaginary components. To subtract two complex numbers \((a + bi)\) and \((c + di)\), follow these steps:

• Subtract the real part of the second number \(c\) from the real part of the first number \(a\), resulting in a new real part: \(a - c\).
• Subtract the imaginary part of the second number \(di\) from the imaginary part of the first number \(bi\), resulting in a new imaginary part: \((bi) - (di) = (b - d)i\).

In the problem, \((12 + 3i) - (7 - 8i)\), we perform these separate subtractions:
1. **Real Parts:** \(12 - 7 = 5\)
2. **Imaginary Parts:** \(3i - (-8i) = 3i + 8i = 11i\)
Combining these gives the result: \(5 + 11i\).
This illustrates that both parts must be handled separately to solve effectively.