Multiplying Complex Numbers and Conjugates

Core Concepts

This stack covers the multiplication of complex numbers, including using the FOIL method. It also introduces the critical concept of the complex conjugate and its property of producing a real number when multiplied by the original complex number.

Multiplying Complex Numbers and Conjugates

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Calculate the product of $4 + 2i$ and its complex conjugate.

The conjugate of $4 + 2i$ is $4 - 2i$. The product is $(4 + 2i)(4 - 2i) = 4^2 + 2^2 = 16 + 4 = 20$.

Calculate: $(5 - 3i)(5 + 3i)$

This is the product of a complex number and its conjugate. The result is $5^2 + 3^2 = 25 + 9 = 34$.

How do you multiply two complex numbers, $(a + bi)$ and $(c + di)$?

Use the FOIL (First, Outer, Inner, Last) method, similar to multiplying binomials. Then, combine like terms and simplify by replacing $i^2$ with $-1$.

$$ (a+bi)(c+di) = ? $$

The product is $ac + adi + bci + bdi^2$. Since $i^2 = -1$, this simplifies to $(ac - bd) + (ad + bc)i$.

Calculate the product: $(3 + 2i)(4 + 5i)$.

FOIL: \\First: $3 \cdot 4 = 12$ \\Outer: $3 \cdot 5i = 15i$ \\Inner: $2i \cdot 4 = 8i$ \\Last: $2i \cdot 5i = 10i^2 = -10$ \\Sum: $12 + 15i + 8i - 10 = 2 + 23i$.

Calculate the product: $(6 - i)(2 - 3i)$.

FOIL: \\First: $6 \cdot 2 = 12$ \\Outer: $6 \cdot (-3i) = -18i$ \\Inner: $-i \cdot 2 = -2i$ \\Last: $-i \cdot (-3i) = 3i^2 = -3$ \\Sum: $12 - 18i - 2i - 3 = 9 - 20i$.

Simplify the expression: $4i(5 - 2i)$.

Use the distributive property: $4i \cdot 5 - 4i \cdot 2i = 20i - 8i^2$. Since $i^2 = -1$, this becomes $20i - 8(-1) = 8 + 20i$.

Simplify the expression $(3+i)^2$.

Expand to $(3+i)(3+i)$. Using FOIL: $9 + 3i + 3i + i^2 = 9 + 6i - 1 = 8 + 6i$.

What is the complex conjugate of a complex number $z = a + bi$?

The complex conjugate, denoted $\bar{z}$, is found by changing the sign of the imaginary part. The conjugate of $a + bi$ is $a - bi$.

Find the complex conjugate of $z = -7 - 3i$.

The complex conjugate $\bar{z}$ is $-7 + 3i$. The real part remains unchanged.

Find the complex conjugate of the purely imaginary number $z = 5i$.

The number can be written as $0 + 5i$. Its conjugate is $0 - 5i$, or simply $-5i$.

What is the result of multiplying a complex number $a + bi$ by its conjugate $a - bi$?

The product is always a non-negative real number.$$ (a + bi)(a - bi) =\\ a^2 - (bi)^2 =\\ a^2 - b^2i^2 =\\ a^2 + b^2 $$

Multiplying Complex Numbers and Conjugates

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