Question

$$(3+4 i)-(2+3 i)$$ Given that \(i=\sqrt{-1}\), what is the value of the expression above? 1\. \(1-i\) 2\. \(1+i\) 3\. \(1+7i\) 4\. \(5+7i\)

Short answer

1 + i (Option 2)

Step-by-step solution

Simplify the Real Parts

First, separately identify and subtract the real parts of the complex numbers. The real part of the first number is 3, and the real part of the second number is 2. The calculation is: 3 - 2 = 1.

Simplify the Imaginary Parts

Next, separately identify and subtract the imaginary parts of the complex numbers. The imaginary part of the first number is 4i, and the imaginary part of the second number is 3i. The calculation is: 4i - 3i = 1i, which can be written as simply i.

Combine the Results

Combine the results from the previous two steps to form the final complex number: 1 + i.

Find the Correct Option

Compare the resultant complex number to the given options. The correct option is: 2. 1 + i.

Key concepts

simplifying complex numbers

Complex numbers are a combination of a real number and an imaginary number. An imaginary number is represented by the symbol \(i\), which is defined as \(i = \sqrt{-1}\). To simplify complex numbers, it is helpful to deal with the real and imaginary parts separately.
In the given exercise, we have two complex numbers: \(3 + 4i\) and \(2 + 3i\). The real parts are 3 and 2, while the imaginary parts are 4i and 3i.
To simplify the given complex numbers, we follow these steps:

• First, subtract the real parts: \(3 - 2 = 1\).
• Then, subtract the imaginary parts: \(4i - 3i = i\).
  

Finally, combine these results to get the simplified complex number: \(1 + i\).
This systematic approach helps in breaking down the complexity and making the simplification process more manageable.

subtracting complex numbers

Subtracting complex numbers is a lot like subtracting vectors. You handle the real and imaginary parts separately, just like you handle horizontal and vertical components in vectors. Let's work through the example given in the exercise.
We start by looking at the two complex numbers: \(3 + 4i\) and \(2 + 3i\). The goal is to subtract the second number from the first.

• First, subtract the real part of the second number from the real part of the first: \(3 - 2 = 1\).
• Next, subtract the imaginary part of the second number from the imaginary part of the first: \(4i - 3i = i\).

So, when you subtract \(2 + 3i\) from \(3 + 4i\), the result is \(1 + i\). This technique ensures you correctly handle each component of the complex numbers, making the subtraction process straightforward and easy to follow.
The key takeaway is to treat real and imaginary parts separately, ensuring a clear and accurate result.

imaginary unit

The imaginary unit, represented by \(i\), is a mathematical concept that allows us to work with numbers that include the square root of negative one. By definition, \(i = \sqrt{-1}\), and it helps in extending the real number system to include complex numbers.
Complex numbers are in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Here, \(b\) is a real number, and \(i\) represents the imaginary unit.
For example, in our exercise, \(3 + 4i\) and \(2 + 3i\) are complex numbers where 3 and 2 are the real parts, and 4i and 3i are the imaginary parts, respectively.

• When simplifying or performing operations on complex numbers, always remember that \(i\) serves to indicate the imaginary part.
• Operations involving \(i\) should be handled carefully, especially in multiplication and division, where unique rules apply due to its definition.

Understanding the imaginary unit \(i\) is crucial for mastering complex numbers. It provides the foundation for more advanced concepts in mathematics and physics.