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Chapter 7: Problem 45

Add or subtract as indicated. Give answers in standard form. $$(-3-4 i)-(-1-4 i)$$

Short Answer

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Step by step solution

01

Identify the terms

The given expression is ewline (-3 - 4i) - (-1 - 4i).
02

Distribute the minus sign

Apply the distributive property to the expression. This changes the subtraction of a negative to addition: ewline (-3 - 4i) + (1 + 4i).
03

Combine like terms

Add the real parts together and the imaginary parts together: ewline (-3 + 1) + (-4i + 4i).
04

Simplify the expression

Perform the necessary operations for the real parts and the imaginary parts: ewline (-3 + 1) = -2 ewline (-4i + 4i) = 0 ewline So, the simplified expression is ewline -2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Addition of Complex Numbers
Addition of complex numbers can seem tricky at first, but it's easier when you break it down. A complex number has two parts: a real part and an imaginary part. The imaginary part is always denoted with an 'i'. To add two complex numbers, you combine their real parts and their imaginary parts separately.
For example, to add (-3 - 4i) and (1 + 4i):
  • First, add the real parts: -3 and +1.
  • Next, add the imaginary parts: -4i and +4i.
This gives you (-3 + 1) + (-4i + 4i) = -2 + 0i. In simpler terms, it means that the complex number becomes -2. The imaginary parts cancel each other out in this case, leaving you with just the real part.
Subtraction of Complex Numbers
Subtraction of complex numbers works much like addition. You treat the real and imaginary parts separately. When you subtract a complex number, you essentially distribute the negative sign to both the real and imaginary parts.
For example, if you have to subtract -(-1 - 4i) from (-3 - 4i):
  • First, rewrite the expression to show the distribution of the negative sign: (-3 - 4i) + (1 + 4i).
  • Next, combine the real parts: -3 and +1.
  • Then, combine the imaginary parts: -4i and +4i.
This gives you the same result as in the addition example: -2 + 0i, which simplifies to -2.
Standard Form of Complex Numbers
The standard form of a complex number is written as (a + bi), where aepresents the real part and biepresents the imaginary part. In any operation involving complex numbers, we aim to express the final result in this standard form.
For the given problem:
After performing the necessary addition or subtraction of real and imaginary parts, we attained the simplified form -2, which is also in the standard form. Here, a = -2db = 0. We write it simply as -2, without the imaginary part, because the imaginary part is zero.
Remember, in the standard form, if bidis 0, you don鈥檛 need to write the imaginary part. Only the real part stays.

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