Problem 86
Question
Simplify each expression and write it in the standard form \(a+b i\). \((6+i)-4 i\)
Step-by-Step Solution
Verified Answer
The expression simplifies to \(6 - 3i\).
1Step 1: Distribute negative sign
When you see a subtraction involving a complex number, it's helpful to distribute the negative sign to each component of the subtracted term. In this expression, you have \((6+i) - 4i\). This simplifies to:\[6 + i - 4i\]
2Step 2: Combine like terms
Now, combine the like terms. The real parts and the imaginary parts should be combined separately. Here, the real part is 6, and the imaginary parts are \(i\) and \(-4i\). Combined, these become:\[6 + (1i - 4i) = 6 - 3i\]
3Step 3: Write in standard form
Once you have combined the like terms, ensure the expression is in standard form \(a + b i\). The simplified expression is:\[6 - 3i\]
Key Concepts
Imaginary UnitReal and Imaginary PartsStandard Form a + bi
Imaginary Unit
In the realm of complex numbers, the imaginary unit is a fundamental concept. It is denoted by the symbol \(i\) and represents the square root of -1, meaning \(i^2 = -1\). This might seem puzzling at first, since negative numbers don't have real square roots in the usual sense. However, the introduction of \(i\) allows us to expand the real number system into the complex number system.
The imaginary unit is the cornerstone of complex numbers, which can be used not only to solve mathematical equations but also to understand more advanced topics such as electrical engineering and quantum mechanics. Here are a few important things to keep in mind:
The imaginary unit is the cornerstone of complex numbers, which can be used not only to solve mathematical equations but also to understand more advanced topics such as electrical engineering and quantum mechanics. Here are a few important things to keep in mind:
- The multiplication of two imaginary units, \(i\) and \(i\), gives -1.
- Multiplying an imaginary unit by itself results in cycling through values: \(i, -1, -i,\) and \(1\), repeating this cycle with successive powers of \(i\).
Real and Imaginary Parts
Every complex number consists of two parts: the real part and the imaginary part. In the expression of a complex number, \(a + bi\), the real part is \(a\), and the imaginary part is \(b\). This forms the building blocks for complex numbers, allowing them to be represented on a two-dimensional plane, known as the complex plane.
The real part is simply a real number, which can be identified as the portion of the complex number that doesn't involve the imaginary unit \(i\). For example, in the complex number \(6 - 3i\), the real part is 6.
The imaginary part, however, is a real number that is multiplied by the imaginary unit \(i\). In our previous example, \(6 - 3i\), the imaginary part is -3. This ensures that the complex number spans across both dimensions of the complex plane. Remember:
The real part is simply a real number, which can be identified as the portion of the complex number that doesn't involve the imaginary unit \(i\). For example, in the complex number \(6 - 3i\), the real part is 6.
The imaginary part, however, is a real number that is multiplied by the imaginary unit \(i\). In our previous example, \(6 - 3i\), the imaginary part is -3. This ensures that the complex number spans across both dimensions of the complex plane. Remember:
- When adding or subtracting complex numbers, combine their real parts and imaginary parts separately.
- The imaginary part includes the coefficient of \(i\), which plays a crucial role in equations.
Standard Form a + bi
The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. This form is essential for simplifying, comparing, and performing operations on complex numbers.
Writing a complex number in standard form involves arranging the number such that the real part \(a\) is written first, followed by the imaginary part \(bi\). This format not only standardizes the expression but also provides an easily recognizable structure for calculations.
Looking at our exercise, simplifying the expression \((6+i) - 4i\) results in \(6 - 3i\), which is already in the standard form. Key points to remember about the standard form include:
Writing a complex number in standard form involves arranging the number such that the real part \(a\) is written first, followed by the imaginary part \(bi\). This format not only standardizes the expression but also provides an easily recognizable structure for calculations.
Looking at our exercise, simplifying the expression \((6+i) - 4i\) results in \(6 - 3i\), which is already in the standard form. Key points to remember about the standard form include:
- The real number \(a\) comes before the imaginary term \(bi\).
- It helps evaluate and simplify arithmetic operations on complex numbers.
- Ensures clarity by separating real and imaginary components.
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