Problem 40 - Precalculus : Building Concepts and Connections

Question

Find \(x+y, x-y, x y,\) and \(x / y\). $$x=2-9 i ; y=-4+6 i$$

Step-by-Step Solution

Answer

The solutions are \(x+y = -2 -3i\), \(x-y = 6 -15i\), \(x y = 46 + 48i\), and \(x / y = 0.88 + 0.46i\).

Step 1: Addition

Add x and y. This is done by completing the operation separately for the real parts and the imaginary parts of the complex numbers. So, \(x+y=(2-9i) + (-4+6i) = (2-4) + (-9i+6i) = -2 -3i \)

Step 2: Subtraction

Subtract y from x. Again, perform the operation separately for the real and imaginary parts. So, \(x-y=(2-9i) - (-4+6i) = (2-(-4)) + (-9i-6i) = 6 -15i \)

Step 3: Multiplication

Multiply x and y. When multiplying complex numbers, use the FOIL (First, Outside, Inside, Last) method from algebra, which applies to binomials (two-term expressions). And remember that \(i^2=-1\). So, \(x y = (2-9i) (-4+6i) = -8 + 12i +36i -54i^2 = -8 + 48i +54 = 46 + 48i\)

Step 4: Division

Divide x by y. Division of complex numbers involves multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number is obtained by changing the sign of its imaginary part. The conjugate of \(y=-4+6i\) is \(-4-6i\). To perform the division, multiply both x and y by this conjugate. So, \(x / y = (2-9i)/(-4+6i) \times (-4-6i)/(-4-6i) = (2-9i)(-4-6i)/(16+36) = -8 -12i + 36i -54i^2/52 = -8 + 24i +54/52 = 46/52 + 24i/52 = 0.88 + 0.46i \)

Key Concepts

Addition of Complex Numbers

Adding complex numbers is all about combining their real and imaginary parts separately. When you look at a complex number, like \(x = 2 - 9i\) and \(y = -4 + 6i\), each has a real part and an imaginary part. To add two complex numbers, do the following:

• Add the real parts: For \(x+y\), we combine \(2\) and \(-4\) to get \(-2\).
• Add the imaginary parts: Combine the \(-9i\) and \(6i\) to get \(-3i\).

Therefore, the sum is \(-2 - 3i\). It’s just like adding regular numbers, but you keep the real and imaginary components separate.

Subtraction of Complex Numbers

Subtraction of complex numbers works similarly to addition. You need to deal with the real and imaginary parts separately again but subtract instead of adding.

• For the real parts: From \(2\), subtract \(-4\), which results in \(6\).
• For the imaginary parts: Subtract \(6i\) from \(-9i\), which gives you \(-15i\).

Thus, the difference, \(x-y\), is \(6 - 15i\). Much like addition, it’s about handling the parts individually but with subtraction.

Multiplication of Complex Numbers

When multiplying complex numbers, use the distributive property (also known as the FOIL method). Applying it to complex numbers can seem tricky at first, but with practice, it becomes second nature.

• First, multiply the real parts: \(2 \times -4 = -8\).
• Outside: \(2 \times 6i = 12i\).
• Inside: \(-9i \times -4 = 36i\).
• Last: \(-9i \times 6i = -54i^2\). Since \(i^2 = -1\), \(-54i^2\) becomes \(54\).

Combine these results: \(-8 + 12i + 36i + 54\), which simplifies to \(46 + 48i\). Multiplying complex numbers allows transformation using algebraic rules with the unique twist of \(i^2 = -1\).

Division of Complex Numbers

Dividing complex numbers involves a bit more work compared to addition, subtraction, or multiplication. The key element here is the conjugate of the denominator. For the complex number \(-4 + 6i\), the conjugate is \(-4 - 6i\). To divide \(x / y = (2-9i)/(-4+6i)\), multiply both the numerator and denominator by the conjugate of the denominator:

• Numerator: The multiplication \((2-9i)(-4-6i) = -8 -12i + 36i -54i^2\), simplifying to \(46 + 24i\).
• Denominator: \((4^2 + 6^2) = 16 + 36 = 52\).

Hence, the division result is \(\frac{46}{52} + \frac{24i}{52}\), which simplifies to approximately \(0.88 + 0.46i\). This technique ensures you remove the imaginary component from the denominator, making your result a cleaner complex number.