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Magazine Chapter 6: Problem 33
Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ \frac{4+3 i}{5-2 i} $$
Short Answer
Expert verified
The simplified complex fraction is \(\frac{14+23i}{29}\). In the form a + bi, we have a = \(\frac{14}{29}\) and b = \(\frac{23}{29}\).
Step by step solution
01
Identify the complex conjugate of the denominator
The complex conjugate of a complex number is found by changing the sign of the imaginary part. The complex conjugate of the denominator 5-2i will be 5+2i.
02
Multiply both the numerator and the denominator by the complex conjugate
We will multiply both the numerator and the denominator of the given complex fraction by the complex conjugate (5+2i) which we found in Step 1:
$$
\frac{4+3i}{5-2i} \times \frac{5+2i}{5+2i}
$$
03
Multiply the numerators
Now, let's multiply the numerators (4+3i) and (5+2i) using the distributive property:
$$
(4+3i)(5+2i) = 4(5) + 4(2i) + 3i(5) + 3i(2i)
$$
This simplifies to:
$$
20 + 8i + 15i + 6i^2
$$
04
Multiply the denominators
Next, let's multiply the denominators (5-2i) and (5+2i) using the difference of squares formula:
$$
(5-2i)(5+2i) = (5)^2 - (2i)^2
$$
This simplifies to:
$$
25 - 4i^2
$$
05
Replace i^2 with -1
Recall that i^2 = -1. Now, let's replace i^2 with -1 in both the numerator and the denominator:
Numerator:
$$
20 + 8i + 15i - 6(1) = 20 + 8i + 15i - 6
$$
Denominator:
$$
25 + 4(1) = 25 + 4
$$
06
Simplify the numerator and the denominator
Now, let's simplify both the numerator and the denominator:
Numerator:
$$
20 - 6 + 8i + 15i = 14 + 23i
$$
Denominator:
$$
25 + 4 = 29
$$
07
Write the complex fraction in the form a + bi
Finally, we have the simplified complex fraction:
$$
\frac{14+23i}{29}
$$
This is already in the form a + bi, where a = 14/29 and b = 23/29.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Conjugate
In complex numbers, the complex conjugate is a fundamental concept. It's a way to simplify expressions, especially when dividing complex numbers. A complex number is generally expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The complex conjugate of \( a + bi \) is \( a - bi \).
This means we only change the sign of the imaginary part.
Multiplying by the conjugate helps to achieve this by making the denominator a real number.
This means we only change the sign of the imaginary part.
- For example, the complex conjugate of \( 5 - 2i \) is \( 5 + 2i \).
- This is useful when you want to eliminate the imaginary unit \( i \) from the denominator of a complex fraction.
Multiplying by the conjugate helps to achieve this by making the denominator a real number.
Imaginary Unit
The imaginary unit is a crucial part of complex numbers, often denoted by \( i \). It is defined by the property \( i^2 = -1 \). This means that \( i \) is not a real number.
Real numbers are numbers you see on the number line, like 2 or -5. Imaginary numbers appear when we take the square root of a negative number, something impossible with just real numbers.
Real numbers are numbers you see on the number line, like 2 or -5. Imaginary numbers appear when we take the square root of a negative number, something impossible with just real numbers.
- For instance, the square root of -1 is \( i \).
- Thus, \( i \) allows us to solve equations that do not have solutions within the realm of real numbers.
Distributive Property
The distributive property is a useful algebraic technique often used in multiplication. It states that a number multiplied by a sum is the same as multiplying each addend separately and then adding the products.
For example, \( a(b + c) = ab + ac \). When working with complex numbers, this property helps in expansions and simplifications.
For example, \( a(b + c) = ab + ac \). When working with complex numbers, this property helps in expansions and simplifications.
- Consider the multiplication of \( (4 + 3i)(5 + 2i) \). Using the distributive property, this expands to \( 4 \times 5 + 4 \times 2i + 3i \times 5 + 3i \times 2i \).
- This step-by-step expansion simplifies the expression and is essential in arithmetic involving complex numbers.
Difference of Squares
The difference of squares is a special pattern in algebra which expresses the idea that two squared values subtracted can be represented as the product of a sum and difference.
It follows the formula \( a^2 - b^2 = (a-b)(a+b) \). This concept is especially helpful when multiplying complex conjugates.
This makes calculations with complex numbers more manageable, particularly when simplifying denominators in fractions.
It follows the formula \( a^2 - b^2 = (a-b)(a+b) \). This concept is especially helpful when multiplying complex conjugates.
- For example, \( (5 - 2i)(5 + 2i) \) simplifies to \( 5^2 - (2i)^2 \).
- Here, \((2i)^2\) simplifies to \(4i^2\), and knowing \( i^2 = -1 \), it becomes \(-4\).
- Thus, \( 25 - 4 = 29 \) yielding a real number, which simplifies the expression.
This makes calculations with complex numbers more manageable, particularly when simplifying denominators in fractions.
