Chapter 19: Problem 5
Combine and simplify. $$\left(\frac{1}{2}+\frac{i}{3}\right)+\left(\frac{1}{4}-\frac{i}{6}\right)$$
Short Answer
Expert verified
\(\frac{3}{4} + \frac{i}{6}\)
Step by step solution
01
Identify Like Terms
Separate the real and imaginary components of the complex numbers. Real terms are those without the imaginary unit 'i', and imaginary terms are those that include the imaginary unit 'i'. For the given complex numbers, \(\frac{1}{2}\) and \(\frac{1}{4}\) are real terms, while \(\frac{i}{3}\) and \(\frac{-i}{6}\) are imaginary terms.
02
Combine Real Parts
Add the real parts of the complex numbers together: \(\frac{1}{2} + \frac{1}{4}\). Find a common denominator if necessary. In this case, the common denominator is 4, so the addition becomes \(\frac{2}{4} + \frac{1}{4}\).
03
Combine Imaginary Parts
Similarly, add the imaginary parts of the complex numbers together: \(\frac{i}{3} + \left(-\frac{i}{6}\right)\). Find a common denominator for these terms, which is 6, and combine them to get \(\frac{2i}{6} - \frac{i}{6}\).
04
Simplify Both Components
Simplify the additions: For the real part, \(\frac{2}{4}+\frac{1}{4} = \frac{3}{4}\). For the imaginary part, \(\frac{2i}{6}-\frac{i}{6} = \frac{i}{6}\).
05
Combine Real and Simplified Imaginary Parts
Write down the final simplified form of the complex number by combining the simplified real and imaginary parts: \(\frac{3}{4} + \frac{i}{6}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Combining Like Terms
In algebra, combining like terms is a fundamental process used for simplifying expressions or equations. This process is just as essential when dealing with complex numbers, which consist of both real and imaginary components. To successfully combine like terms involving complex numbers, first separate the real from the imaginary parts. Real numbers are considered 'like' with other real numbers, and the same goes for imaginary numbers.
When you add or subtract complex numbers, always combine the real numbers with real numbers and combine the imaginary parts with each other, taking care to include the imaginary unit 'i' with its respective coefficients. In the exercise \( \left(\frac{1}{2}+\frac{i}{3}\right) + \left(\frac{1}{4}-\frac{i}{6}\right) \), the real parts \( \frac{1}{2} \) and \( \frac{1}{4} \) are like terms, as are the imaginary parts \( \frac{i}{3} \) and \( -\frac{i}{6} \). By keeping like terms together, simplification becomes a straightforward process.
When you add or subtract complex numbers, always combine the real numbers with real numbers and combine the imaginary parts with each other, taking care to include the imaginary unit 'i' with its respective coefficients. In the exercise \( \left(\frac{1}{2}+\frac{i}{3}\right) + \left(\frac{1}{4}-\frac{i}{6}\right) \), the real parts \( \frac{1}{2} \) and \( \frac{1}{4} \) are like terms, as are the imaginary parts \( \frac{i}{3} \) and \( -\frac{i}{6} \). By keeping like terms together, simplification becomes a straightforward process.
Complex Arithmetic
Performing arithmetic on complex numbers adheres to the same principles as arithmetic on real numbers, but with an added layer of attention paid to the imaginary unit 'i'. Complex arithmetic involves the addition, subtraction, multiplication, division, and exponentiation of complex numbers.
In the given exercise, the operation of interest is addition. To navigate through complex addition, remember that you simply need to add the real components together and the imaginary components together separately. Then, reassess your resulting expressions to make sure that they are in the simplest form. This often involves finding a common denominator for fractions, just as seen in the example where the denominators were harmonized to 4 and 6 for the real and imaginary parts, respectively. Careful and methodical arithmetic operations ensure that complex numbers are combined and simplified correctly.
In the given exercise, the operation of interest is addition. To navigate through complex addition, remember that you simply need to add the real components together and the imaginary components together separately. Then, reassess your resulting expressions to make sure that they are in the simplest form. This often involves finding a common denominator for fractions, just as seen in the example where the denominators were harmonized to 4 and 6 for the real and imaginary parts, respectively. Careful and methodical arithmetic operations ensure that complex numbers are combined and simplified correctly.
Imaginary Unit Manipulation
The imaginary unit, denoted as 'i', is the fundamental building block of imaginary and complex numbers. It is defined by the property that \( i^2 = -1 \). Manipulating the imaginary unit correctly is crucial in complex number arithmetic.
When combining complex numbers, as in \( \frac{i}{3} + \left(-\frac{i}{6}\right) \), it’s important to treat 'i' as a variable or algebraic placeholder that signifies which terms are imaginary. However, unlike other algebraic variables, 'i' has a special property that allows for further simplification when raised to higher powers (for example, \( i^3 = i^2\cdot i = -1\cdot i = -i \) and \( i^4 = i^2\cdot i^2 = (-1)\cdot (-1) = 1 \)). Always check if you can simplify expressions with higher powers of 'i' using these properties. For addition and subtraction, keep 'i' as part of the coefficient for clarity and consistency.
When combining complex numbers, as in \( \frac{i}{3} + \left(-\frac{i}{6}\right) \), it’s important to treat 'i' as a variable or algebraic placeholder that signifies which terms are imaginary. However, unlike other algebraic variables, 'i' has a special property that allows for further simplification when raised to higher powers (for example, \( i^3 = i^2\cdot i = -1\cdot i = -i \) and \( i^4 = i^2\cdot i^2 = (-1)\cdot (-1) = 1 \)). Always check if you can simplify expressions with higher powers of 'i' using these properties. For addition and subtraction, keep 'i' as part of the coefficient for clarity and consistency.
