Problem 78
Question
Use the following definition. A complex number \(a+b i\) is often denoted by the letter \(z .\) Its conjugate, \(a-b i\) is denoted by \(\bar{z}\). Show that \(z \bar{z}=a^{2}+b^{2}\)
Step-by-Step Solution
Verified Answer
The multiplication of a complex number and its conjugate is indeed equal to the sum of squares of the components of the complex number.
1Step 1: Understand the Definitions
A complex number is represented in the form \(a+bi\) where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. The conjugate of this complex number is given by \(a-bi\). This is denoted as \(\bar z\).
2Step 2: Multiply the Complex Number by its Conjugate
Multiply the complex number \(z\) by its conjugate \(\bar z\). This results in \((a+bi)\) times \((a-bi)\). Now apply the rule for multiplication of binomials, which states \((x+y)(x-y)=x^{2}-y^{2}\). After applying this rule, we get \(a^{2} - (bi)^{2}\).
3Step 3: Simplify the Equation
On simplifying the above expression, we get \(a^{2} - b^{2}i^{2}\). Since \(i^{2} = -1\), we obtain \(a^{2} + b^{2}\). Hence it's proven that \(z \bar z=a^{2}+b^{2}\).
Key Concepts
Complex ConjugateMultiplication of BinomialsImaginary UnitReal Numbers
Complex Conjugate
In the realm of complex numbers, a significant concept is that of the complex conjugate. If you have a complex number denoted as \( z = a + bi \), its conjugate, represented as \( \bar{z} \), is \( a - bi \). The complex conjugate is essentially the same as the original number, but with the sign of the imaginary part changed.
This reversal is crucial when simplifying products involving complex numbers. It's a fundamental tool in mathematics that helps in reducing expressions to just real numbers. Why does this matter? Because multiplying a complex number by its conjugate will eliminate the imaginary part altogether.
When you multiply \( z \) by \( \bar{z} \), you end up with a real number. This property is incredibly useful in complex number division, allowing us to "clear" the imaginary components and simplify calculations.
This reversal is crucial when simplifying products involving complex numbers. It's a fundamental tool in mathematics that helps in reducing expressions to just real numbers. Why does this matter? Because multiplying a complex number by its conjugate will eliminate the imaginary part altogether.
When you multiply \( z \) by \( \bar{z} \), you end up with a real number. This property is incredibly useful in complex number division, allowing us to "clear" the imaginary components and simplify calculations.
Multiplication of Binomials
To multiply complex numbers like \( z = a + bi \) and \( \bar{z} = a - bi \), you utilize the multiplication of binomials. The formula applied here is \((x+y)(x-y) = x^2 - y^2\). In these cases, A common technique in algebra is used, known as making use of the difference of squares.
By applying this formula, you calculate \((a + bi)(a - bi)\). Here, \(x\) is equivalent to \(a\) and \(y\) to \(bi\). Following the formula gives \(a^2 - (bi)^2\). This calculation beautifully demonstrates how imaginary units simplify when paired with their conjugate.
Ultimately, understanding the multiplication of binomials empowers us to simplify complex number expressions accurately and efficiently.
By applying this formula, you calculate \((a + bi)(a - bi)\). Here, \(x\) is equivalent to \(a\) and \(y\) to \(bi\). Following the formula gives \(a^2 - (bi)^2\). This calculation beautifully demonstrates how imaginary units simplify when paired with their conjugate.
Ultimately, understanding the multiplication of binomials empowers us to simplify complex number expressions accurately and efficiently.
Imaginary Unit
The imaginary unit, denoted by \(i\), is a fundamental component of complex numbers. It commands a simple yet powerful rule: \(i^2 = -1\). This property is what sets the stage for much of complex number arithmetic, including the problem at hand.
Complex numbers extend real numbers by including this unit, allowing computations that are otherwise impossible with just real numbers. When dealing with complex numbers and their conjugates, it is crucial to remember this property.
In expressions like \( a^2 - b^2i^2 \), the term \(-b^2i^2\) simplifies to \(b^2\) because \(i^2\) is indeed \(-1\). It is this simplification that yields a real number from a potentially complex expression.
Complex numbers extend real numbers by including this unit, allowing computations that are otherwise impossible with just real numbers. When dealing with complex numbers and their conjugates, it is crucial to remember this property.
In expressions like \( a^2 - b^2i^2 \), the term \(-b^2i^2\) simplifies to \(b^2\) because \(i^2\) is indeed \(-1\). It is this simplification that yields a real number from a potentially complex expression.
Real Numbers
Real numbers are familiar territory for many. They include both integers and fractions, encompassing all numbers we count, measure, and calculate in everyday life.
In the context of complex numbers, real numbers serve as the groundwork. When multiplying a complex number by its conjugate, like in \( z \bar{z} \), the result we aim for is entirely real—\(a^2 + b^2\). Here, no imaginary parts remain.
At its core, the operation of multiplying complex conjugates underscores the interplay between complex numbers and real numbers. By forcing the imaginary to disappear, we rediscover a facet of mathematics that circles back to the real realm, making complex numbers not so complex after all.
In the context of complex numbers, real numbers serve as the groundwork. When multiplying a complex number by its conjugate, like in \( z \bar{z} \), the result we aim for is entirely real—\(a^2 + b^2\). Here, no imaginary parts remain.
At its core, the operation of multiplying complex conjugates underscores the interplay between complex numbers and real numbers. By forcing the imaginary to disappear, we rediscover a facet of mathematics that circles back to the real realm, making complex numbers not so complex after all.
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