Complex Numbers: Detailed Explanation and 100 Examples

Introduction to Complex Numbers

A complex number is a number that can be written in the form:

$z=a+bi$

Where:

• $a$ is the real part,
• $b$ is the imaginary part, and
• $i$ is the imaginary unit, defined as $i^2=-1$.

Complex numbers extend the idea of real numbers and are used to solve equations that do not have real solutions. For example, the equation $x^2+1=0$ has no real solution, but in the system of complex numbers, we can express the solution as $x=i$ and $x=-i$.

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Example 1: Adding Complex Numbers

Given two complex numbers $z_1=3+4i$ and $z_2=1–2i$, find their sum.

Solution:

Step 1: Add the real parts:
$3+1=4$

Step 2: Add the imaginary parts:
$4i+(-2i)=2i$

Thus, the sum is:
$z_1+z_2=4+2i$

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Example 2: Subtracting Complex Numbers

Given two complex numbers $z_1=5+7i$ and $z_2=2–3i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$5–2=3$

Step 2: Subtract the imaginary parts:
$7i–(-3i)=7i+3i=10i$

Thus, the difference is:
$z_1–z_2=3+10i$

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Example 3: Multiplying Complex Numbers

Given two complex numbers $z_1=2+3i$ and $z_2=4–i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(2+3i)(4–i)=2(4)+2(-i)+3i(4)+3i(-i)$

Step 2: Simplify each term:
$2(4)=8$
$2(-i)=-2i$
$3i(4)=12i$
$3i(-i)=-3i^2$

Step 3: Recall that $i^2=-1$, so:
$-3i^2=3$

Step 4: Combine all terms:
$8–2i+12i+3=11+10i$

Thus, the product is:
$z_1\times z_2=11+10i$

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Example 4: Dividing Complex Numbers

Given two complex numbers $z_1=6+5i$ and $z_2=2–i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator $z_2=2–i$:
$\frac{6+5i}{2–i}\times \frac{2+i}{2+i}=\frac{(6+5i)(2+i)}{(2–i)(2+i)}$

Step 2: Use the distributive property in the numerator:
$=(6)(2)+(6)(i)+(5i)(2)+(5i)(i)$

Step 3: Simplify each term:
$6(2)=12$
$6(i)=6i$
$5i(2)=10i$
$5i(i)=5i^2$

Step 4: Recall that $i^2=-1$, so:
$5i^2=-5$

Step 5: Combine the terms in the numerator:
$12+6i+10i–5=7+16i$

Step 6: Simplify the denominator using the difference of squares formula:
$(2–i)(2+i)=2^2–i^2=4–(-1)=5$

Step 7: Write the final quotient:
$\frac{7+16i}{5}=\frac{7}{5}+\frac{16}{5}i$

Thus, the quotient is:
$z_1÷z_2=\frac{7}{5}+\frac{16}{5}i$

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Example 5: Finding the Modulus of a Complex Number

Given a complex number $z=3+4i$, find its modulus.

Solution:

Step 1: The modulus of a complex number $z=a+bi$ is given by:
$|z|=\sqrt{a^2+b^2}$

Step 2: Square the real part:
$3^2=9$

Step 3: Square the imaginary part:
$4^2=16$

Step 4: Add the squares:
$9+16=25$

Step 5: Take the square root:
$\sqrt{25}=5$

Thus, the modulus is:
$|z|=5$

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Example 6: Conjugate of a Complex Number

Given a complex number $z=5–7i$, find its conjugate.

Solution:

Step 1: The conjugate of $z=a+bi$ is given by:
$\bar{z}=a–bi$

Step 2: For $z=5–7i$, the conjugate is:
$\bar{z}=5+7i$

Thus, the conjugate is:
$\bar{z}=5+7i$

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Example 7: Adding Conjugates

Given two complex numbers $z_1=4+5i$ and $z_2=4–5i$, find their sum.

Solution:

Step 1: Add the real parts:
$4+4=8$

Step 2: Add the imaginary parts:
$5i+(-5i)=0$

Thus, the sum is:
$z_1+z_2=8$

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Example 8: Subtracting Conjugates

Given two complex numbers $z_1=3+2i$ and $z_2=3–2i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$3–3=0$

Step 2: Subtract the imaginary parts:
$2i–(-2i)=2i+2i=4i$

Thus, the difference is:
$z_1–z_2=4i$

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Example 9: Modulus of a Conjugate

Given a complex number $z=6+8i$, find the modulus of its conjugate $\bar{z}$.

Solution:

Step 1: The conjugate of $z=6+8i$ is $\bar{z}=6–8i$.

Step 2: The modulus of $\bar{z}=6–8i$ is the same as the modulus of $z$:
$|z|=|6+8i|$

Step 3: Use the formula for modulus:
$|z|=\sqrt{6^2+8^2}$

Step 4: Square the real part:
$6^2=36$

Step 5: Square the imaginary part:
$8^2=64$

Step 6: Add the squares:
$36+64=100$

Step 7: Take the square root:
$\sqrt{100}=10$

Thus, the modulus is:
$|\bar{z}|=10$

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Example 10: Multiplying Conjugates

Given two complex numbers $z_1=4+3i$ and $z_2=4–3i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(4+3i)(4–3i)$
$=4(4)+4(-3i)+3i(4)+3i(-3i)$

Step 2: Simplify each term:
$4(4)=16$
$4(-3i)=-12i$
$3i(4)=12i$
$3i(-3i)=-9i^2$

Step 3: Recall that $i^2=-1$:
$-9i^2$
$=9$

Step 4: Combine like terms:
$16–12i+12i+9$
$=25$

Thus, the product is:
$z_1\times z_2$
$=25$

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Example 11: Dividing Conjugates

Given two complex numbers $z_1=6+2i$ and $z_2=6–2i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{6+2i}{6–2i}\times \frac{6+2i}{6+2i}$
$=\frac{(6+2i)(6+2i)}{(6–2i)(6+2i)}$

Step 2: Use the distributive property (FOIL) in the numerator:
$(6+2i)(6+2i)$
$=6(6)+6(2i)+2i(6)+2i(2i)$

Step 3: Simplify each term:
$6(6)=36$
$6(2i)=12i$
$2i(6)=12i$
$2i(2i)=4i^2$

Step 4: Recall that $i^2=-1$:
$4i^2$
$=-4$

Step 5: Combine terms in the numerator:
$36+12i+12i–4$
$=32+24i$

Step 6: Use the difference of squares formula in the denominator:
$(6–2i)(6+2i)$
$=6^2–(2i{)}^2$
$=36–(-4)$
$=40$

Step 7: Write the final quotient:
$\frac{32+24i}{40}$
$=\frac{32}{40}+\frac{24i}{40}$
$=\frac{4}{5}+\frac{3i}{5}$

Thus, the quotient is:
$z_1÷z_2$
$=\frac{4}{5}+\frac{3}{5}i$

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Example 12: Finding the Modulus of a Complex Number

Given a complex number $z=7–24i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=7$
$b=-24$

Step 3: Square the real part:
$7^2$
$=49$

Step 4: Square the imaginary part:
$(-24{)}^2$
$=576$

Step 5: Add the squares:
$49+576$
$=625$

Step 6: Take the square root:
$\sqrt{625}$
$=25$

Thus, the modulus is:
$|z|=25$

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Example 13: Conjugate of a Complex Number

Given a complex number $z=9+10i$, find its conjugate.

Solution:

Step 1: The conjugate of $z=a+bi$ is given by:
$\bar{z}=a–bi$

Step 2: Identify the real and imaginary parts:
$a=9$
$b=10$

Step 3: Write the conjugate:
$\bar{z}=9–10i$

Thus, the conjugate is:
$\bar{z}=9–10i$

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Example 14: Adding Complex Numbers

Given two complex numbers $z_1=3+4i$ and $z_2=6+8i$, find their sum.

Solution:

Step 1: Add the real parts:
$3+6=9$

Step 2: Add the imaginary parts:
$4i+8i=12i$

Thus, the sum is:
$z_1+z_2=9+12i$

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Example 15: Subtracting Complex Numbers

Given two complex numbers $z_1=7+9i$ and $z_2=2+3i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$7–2=5$

Step 2: Subtract the imaginary parts:
$9i–3i=6i$

Thus, the difference is:
$z_1–z_2=5+6i$

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Example 16: Multiplying Complex Numbers

Given two complex numbers $z_1=3+2i$ and $z_2=1+4i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(3+2i)(1+4i)$
$=3(1)+3(4i)+2i(1)+2i(4i)$

Step 2: Simplify each term:
$3(1)=3$
$3(4i)=12i$
$2i(1)=2i$
$2i(4i)=8i^2$

Step 3: Recall that $i^2=-1$:
$8i^2$
$=-8$

Step 4: Combine all terms:
$3+12i+2i–8$
$=-5+14i$

Thus, the product is:
$z_1\times z_2$
$=-5+14i$

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Example 17: Dividing Complex Numbers

Given two complex numbers $z_1=8+3i$ and $z_2=4–i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{8+3i}{4–i}\times \frac{4+i}{4+i}$
$=\frac{(8+3i)(4+i)}{(4–i)(4+i)}$

Step 2: Use the distributive property (FOIL) in the numerator:
$(8+3i)(4+i)$
$=8(4)+8(i)+3i(4)+3i(i)$

Step 3: Simplify each term:
$8(4)=32$
$8(i)=8i$
$3i(4)=12i$
$3i(i)=3i^2$

Step 4: Recall that $i^2=-1$:
$3i^2$
$=-3$

Step 5: Combine the terms in the numerator:
$32+8i+12i–3$
$=29+20i$

Step 6: Simplify the denominator using the difference of squares formula:
$(4–i)(4+i)$
$=4^2–i^2$
$=16–(-1)$
$=17$

Step 7: Write the final quotient:
$\frac{29+20i}{17}$
$=\frac{29}{17}+\frac{20i}{17}$

Thus, the quotient is:
$z_1÷z_2$
$=\frac{29}{17}+\frac{20i}{17}$

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Example 18: Dividing Complex Numbers

Given two complex numbers $z_1=5+2i$ and $z_2=3–i$, find their quotient.

Solution:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator:
$\frac{5+2i}{3–i}\times \frac{3+i}{3+i}$
$=\frac{(5+2i)(3+i)}{(3–i)(3+i)}$

Step 2: Use the distributive property (FOIL) in the numerator:
$(5+2i)(3+i)$
$=5(3)+5(i)+2i(3)+2i(i)$

Step 3: Simplify each term:
$5(3)=15$
$5(i)=5i$
$2i(3)=6i$
$2i(i)=2i^2$

Step 4: Recall that $i^2=-1$:
$2i^2=-2$

Step 5: Combine the terms in the numerator:
$15+5i+6i–2$
$=13+11i$

Step 6: Use the difference of squares formula in the denominator:
$(3–i)(3+i)$
$=3^2–i^2$
$=9–(-1)$
$=10$

Step 7: Write the final quotient:
$\frac{13+11i}{10}$
$=\frac{13}{10}+\frac{11i}{10}$

Thus, the quotient is:
$z_1÷z_2$
$=\frac{13}{10}+\frac{11i}{10}$

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Example 19: Finding the Modulus of a Complex Number

Given a complex number $z=9+12i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=9$
$b=12$

Step 3: Square the real part:
$9^2=81$

Step 4: Square the imaginary part:
${12}^2=144$

Step 5: Add the squares:
$81+144=225$

Step 6: Take the square root:
$\sqrt{225}=15$

Thus, the modulus is:
$|z|=15$

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Example 20: Adding Complex Numbers

Given two complex numbers $z_1=6+5i$ and $z_2=2+3i$, find their sum.

Solution:

Step 1: Add the real parts:
$6+2=8$

Step 2: Add the imaginary parts:
$5i+3i=8i$

Thus, the sum is:
$z_1+z_2=8+8i$

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Example 21: Subtracting Complex Numbers

Given two complex numbers $z_1=10+7i$ and $z_2=4+2i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$10–4=6$

Step 2: Subtract the imaginary parts:
$7i–2i=5i$

Thus, the difference is:
$z_1–z_2=6+5i$

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Example 22: Multiplying Complex Numbers

Given two complex numbers $z_1=3+4i$ and $z_2=1+2i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(3+4i)(1+2i)$
$=3(1)+3(2i)+4i(1)+4i(2i)$

Step 2: Simplify each term:
$3(1)=3$
$3(2i)=6i$
$4i(1)=4i$
$4i(2i)=8i^2$

Step 3: Recall that $i^2=-1$:
$8i^2=-8$

Step 4: Combine all terms:
$3+6i+4i–8$
$=-5+10i$

Thus, the product is:
$z_1\times z_2=-5+10i$

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Example 23: Dividing Complex Numbers

Given two complex numbers $z_1=7+9i$ and $z_2=2+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator:
$\frac{7+9i}{2+i}\times \frac{2–i}{2–i}$
$=\frac{(7+9i)(2–i)}{(2+i)(2–i)}$

Step 2: Use the distributive property (FOIL) in the numerator:
$(7+9i)(2–i)$
$=7(2)+7(-i)+9i(2)+9i(-i)$

Step 3: Simplify each term:
$7(2)=14$
$7(-i)=-7i$
$9i(2)=18i$
$9i(-i)=-9i^2$

Step 4: Recall that $i^2=-1$:
$-9i^2=9$

Step 5: Combine the terms in the numerator:
$14–7i+18i+9$
$=23+11i$

Step 6: Use the difference of squares formula in the denominator:
$(2+i)(2–i)=2^2–i^2$
$=4–(-1)$
$=5$

Step 7: Write the final quotient:
$\frac{23+11i}{5}$
$=\frac{23}{5}+\frac{11i}{5}$

Thus, the quotient is:
$z_1÷z_2=\frac{23}{5}+\frac{11i}{5}$

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Example 24: Dividing Complex Numbers

Given two complex numbers $z_1=5+6i$ and $z_2=3+2i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{5+6i}{3+2i}\times \frac{3–2i}{3–2i}$
$=\frac{(5+6i)(3–2i)}{(3+2i)(3–2i)}$

Step 2: Use the distributive property (FOIL):
$(5+6i)(3–2i)$
$=5(3)+5(-2i)+6i(3)+6i(-2i)$

Step 3: Simplify each term:
$5(3)=15$
$5(-2i)=-10i$
$6i(3)=18i$
$6i(-2i)=-12i^2$

Step 4: Recall that $i^2=-1$:
$-12i^2$
$=12$

Step 5: Combine the terms in the numerator:
$15–10i+18i+12$
$=27+8i$

Step 6: Use the difference of squares formula in the denominator:
$(3+2i)(3–2i)=3^2–(2i{)}^2$
$=9–(-4)$
$=13$

Step 7: Write the final quotient:
$\frac{27+8i}{13}$
$=\frac{27}{13}+\frac{8i}{13}$

Thus, the quotient is:
$z_1÷z_2$
$=\frac{27}{13}+\frac{8i}{13}$

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Example 25: Multiplying Complex Numbers

Given two complex numbers $z_1=4+5i$ and $z_2=2–3i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(4+5i)(2–3i)$
$=4(2)+4(-3i)+5i(2)+5i(-3i)$

Step 2: Simplify each term:
$4(2)=8$
$4(-3i)=-12i$
$5i(2)=10i$
$5i(-3i)=-15i^2$

Step 3: Recall that $i^2=-1$:
$-15i^2$
$=15$

Step 4: Combine all terms:
$8–12i+10i+15$
$=23–2i$

Thus, the product is:
$z_1\times z_2$
$=23–2i$

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Example 26: Finding the Modulus of a Complex Number

Given a complex number $z=8+15i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=8$
$b=15$

Step 3: Square the real part:
$8^2=64$

Step 4: Square the imaginary part:
${15}^2=225$

Step 5: Add the squares:
$64+225=289$

Step 6: Take the square root:
$\sqrt{289}=17$

Thus, the modulus is:
$|z|=17$

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Example 27: Adding Complex Numbers

Given two complex numbers $z_1=9+4i$ and $z_2=7+3i$, find their sum.

Solution:

Step 1: Add the real parts:
$9+7=16$

Step 2: Add the imaginary parts:
$4i+3i=7i$

Thus, the sum is:
$z_1+z_2=16+7i$

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Example 28: Subtracting Complex Numbers

Given two complex numbers $z_1=11+8i$ and $z_2=5+6i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$11–5=6$

Step 2: Subtract the imaginary parts:
$8i–6i=2i$

Thus, the difference is:
$z_1–z_2=6+2i$

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Example 29: Dividing Complex Numbers

Given two complex numbers $z_1=4+7i$ and $z_2=1–2i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{4+7i}{1–2i}\times \frac{1+2i}{1+2i}$
$=\frac{(4+7i)(1+2i)}{(1–2i)(1+2i)}$

Step 2: Use the distributive property (FOIL):
$(4+7i)(1+2i)$
$=4(1)+4(2i)+7i(1)+7i(2i)$

Step 3: Simplify each term:
$4(1)=4$
$4(2i)=8i$
$7i(1)=7i$
$7i(2i)=14i^2$

Step 4: Recall that $i^2=-1$:
$14i^2=-14$

Step 5: Combine the terms in the numerator:
$4+8i+7i–14=-10+15i$

Step 6: Use the difference of squares formula in the denominator:
$(1–2i)(1+2i)=1^2–(2i{)}^2$
$=1–(-4)=5$

Step 7: Write the final quotient:
$\frac{-10+15i}{5}=\frac{-10}{5}+\frac{15i}{5}$
$=-2+3i$

Thus, the quotient is:
$z_1÷z_2=-2+3i$

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Example 30: Multiplying Complex Numbers

Given two complex numbers $z_1=2+3i$ and $z_2=1–4i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(2+3i)(1–4i)$
$=2(1)+2(-4i)+3i(1)+3i(-4i)$

Step 2: Simplify each term:
$2(1)=2$
$2(-4i)=-8i$
$3i(1)=3i$
$3i(-4i)=-12i^2$

Step 3: Recall that $i^2=-1$:
$-12i^2=12$

Step 4: Combine all terms:
$2–8i+3i+12$
$=14–5i$

Thus, the product is:
$z_1\times z_2=14–5i$

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Example 31: Finding the Modulus of a Complex Number

Given a complex number $z=6+8i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=6$
$b=8$

Step 3: Square the real part:
$6^2=36$

Step 4: Square the imaginary part:
$8^2=64$

Step 5: Add the squares:
$36+64=100$

Step 6: Take the square root:
$\sqrt{100}=10$

Thus, the modulus is:
$|z|=10$

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Example 32: Adding Complex Numbers

Given two complex numbers $z_1=7+4i$ and $z_2=2+3i$, find their sum.

Solution:

Step 1: Add the real parts:
$7+2=9$

Step 2: Add the imaginary parts:
$4i+3i=7i$

Thus, the sum is:
$z_1+z_2=9+7i$

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Example 33: Subtracting Complex Numbers

Given two complex numbers $z_1=12+9i$ and $z_2=5+4i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$12–5=7$

Step 2: Subtract the imaginary parts:
$9i–4i=5i$

Thus, the difference is:
$z_1–z_2=7+5i$

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Example 34: Dividing Complex Numbers

Given two complex numbers $z_1=8+6i$ and $z_2=2+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{8+6i}{2+i}\times \frac{2–i}{2–i}$
$=\frac{(8+6i)(2–i)}{(2+i)(2–i)}$

Step 2: Use the distributive property (FOIL):
$(8+6i)(2–i)$
$=8(2)+8(-i)+6i(2)+6i(-i)$

Step 3: Simplify each term:
$8(2)=16$
$8(-i)=-8i$
$6i(2)=12i$
$6i(-i)=-6i^2$

Step 4: Recall that $i^2=-1$:
$-6i^2=6$

Step 5: Combine the terms in the numerator:
$16–8i+12i+6=22+4i$

Step 6: Use the difference of squares formula in the denominator:
$(2+i)(2–i)=2^2–i^2$
$=4–(-1)=5$

Step 7: Write the final quotient:
$\frac{22+4i}{5}=\frac{22}{5}+\frac{4i}{5}$

Thus, the quotient is:
$z_1÷z_2=\frac{22}{5}+\frac{4i}{5}$

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Example 35: Multiplying Complex Numbers

Given two complex numbers $z_1=3+4i$ and $z_2=2–5i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(3+4i)(2–5i)$
$=3(2)+3(-5i)+4i(2)+4i(-5i)$

Step 2: Simplify each term:
$3(2)=6$
$3(-5i)=-15i$
$4i(2)=8i$
$4i(-5i)=-20i^2$

Step 3: Recall that $i^2=-1$:
$-20i^2$
$=20$

Step 4: Combine like terms:
$6–15i+8i+20$
$=26–7i$

Thus, the product is:
$z_1\times z_2=26–7i$

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Example 36: Dividing Complex Numbers

Given two complex numbers $z_1=6+9i$ and $z_2=3–i$, find their quotient.

Solution:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator:
$\frac{6+9i}{3–i}\times \frac{3+i}{3+i}$
$=\frac{(6+9i)(3+i)}{(3–i)(3+i)}$

Step 2: Use the distributive property (FOIL):
$(6+9i)(3+i)$
$=6(3)+6(i)+9i(3)+9i(i)$

Step 3: Simplify each term:
$6(3)=18$
$6(i)=6i$
$9i(3)=27i$
$9i(i)=9i^2$

Step 4: Recall that $i^2=-1$:
$9i^2=-9$

Step 5: Combine the terms in the numerator:
$18+6i+27i–9$
$=9+33i$

Step 6: Use the difference of squares formula in the denominator:
$(3–i)(3+i)=3^2–i^2$
$=9–(-1)=10$

Step 7: Write the final quotient:
$\frac{9+33i}{10}=\frac{9}{10}+\frac{33i}{10}$

Thus, the quotient is:
$z_1÷z_2=\frac{9}{10}+\frac{33i}{10}$

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Example 37: Finding the Modulus of a Complex Number

Given a complex number $z=1+5i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=1$
$b=5$

Step 3: Square the real part:
$1^2=1$

Step 4: Square the imaginary part:
$5^2=25$

Step 5: Add the squares:
$1+25=26$

Step 6: Take the square root:
$\sqrt{26}$

Thus, the modulus is:
$|z|=\sqrt{26}$

---

Example 38: Adding Complex Numbers

Given two complex numbers $z_1=7+5i$ and $z_2=9+2i$, find their sum.

Solution:

Step 1: Add the real parts:
$7+9=16$

Step 2: Add the imaginary parts:
$5i+2i=7i$

Thus, the sum is:
$z_1+z_2=16+7i$

---

Example 39: Subtracting Complex Numbers

Given two complex numbers $z_1=11+6i$ and $z_2=4+3i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$11–4=7$

Step 2: Subtract the imaginary parts:
$6i–3i=3i$

Thus, the difference is:
$z_1–z_2=7+3i$

---

Example 40: Dividing Complex Numbers

Given two complex numbers $z_1=10+8i$ and $z_2=2–i$, find their quotient.

Solution:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator:
$\frac{10+8i}{2–i}\times \frac{2+i}{2+i}$
$=\frac{(10+8i)(2+i)}{(2–i)(2+i)}$

Step 2: Use the distributive property (FOIL):
$(10+8i)(2+i)$
$=10(2)+10(i)+8i(2)+8i(i)$

Step 3: Simplify each term:
$10(2)=20$
$10(i)=10i$
$8i(2)=16i$
$8i(i)=8i^2$

Step 4: Recall that $i^2=-1$:
$8i^2=-8$

Step 5: Combine the terms in the numerator:
$20+10i+16i–8$
$=12+26i$

Step 6: Use the difference of squares formula in the denominator:
$(2–i)(2+i)=2^2–i^2$
$=4–(-1)=5$

Step 7: Write the final quotient:
$\frac{12+26i}{5}=\frac{12}{5}+\frac{26i}{5}$

Thus, the quotient is:
$z_1÷z_2=\frac{12}{5}+\frac{26i}{5}$

---

Example 41: Multiplying Complex Numbers

Given two complex numbers $z_1=5+6i$ and $z_2=3–2i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(5+6i)(3–2i)$
$=5(3)+5(-2i)+6i(3)+6i(-2i)$

Step 2: Simplify each term:
$5(3)=15$
$5(-2i)=-10i$
$6i(3)=18i$
$6i(-2i)=-12i^2$

Step 3: Recall that $i^2=-1$:
$-12i^2$
$=12$

Step 4: Combine like terms:
$15–10i+18i+12$
$=27+8i$

Thus, the product is:
$z_1\times z_2$
$=27+8i$

---

Example 42: Dividing Complex Numbers

Given two complex numbers $z_1=7+9i$ and $z_2=2+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator:
$\frac{7+9i}{2+i}\times \frac{2–i}{2–i}$
$=\frac{(7+9i)(2–i)}{(2+i)(2–i)}$

Step 2: Use the distributive property (FOIL):
$(7+9i)(2–i)$
$=7(2)+7(-i)+9i(2)+9i(-i)$

Step 3: Simplify each term:
$7(2)=14$
$7(-i)=-7i$
$9i(2)=18i$
$9i(-i)=-9i^2$

Step 4: Recall that $i^2=-1$:
$-9i^2$
$=9$

Step 5: Combine the terms in the numerator:
$14–7i+18i+9$
$=23+11i$

Step 6: Use the difference of squares formula in the denominator:
$(2+i)(2–i)=2^2–i^2$
$=4–(-1)=5$

Step 7: Write the final quotient:
$\frac{23+11i}{5}=\frac{23}{5}+\frac{11i}{5}$

Thus, the quotient is:
$z_1÷z_2=\frac{23}{5}+\frac{11i}{5}$

---

Example 43: Finding the Modulus of a Complex Number

Given a complex number $z=3+4i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=3$
$b=4$

Step 3: Square the real part:
$3^2=9$

Step 4: Square the imaginary part:
$4^2=16$

Step 5: Add the squares:
$9+16=25$

Step 6: Take the square root:
$\sqrt{25}=5$

Thus, the modulus is:
$|z|=5$

---

Example 44: Adding Complex Numbers

Given two complex numbers $z_1=6+2i$ and $z_2=4+3i$, find their sum.

Solution:

Step 1: Add the real parts:
$6+4=10$

Step 2: Add the imaginary parts:
$2i+3i=5i$

Thus, the sum is:
$z_1+z_2=10+5i$

---

Example 45: Subtracting Complex Numbers

Given two complex numbers $z_1=9+5i$ and $z_2=3+2i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$9–3=6$

Step 2: Subtract the imaginary parts:
$5i–2i=3i$

Thus, the difference is:
$z_1–z_2=6+3i$

---

Example 46: Dividing Complex Numbers

Given two complex numbers $z_1=5+7i$ and $z_2=3+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator:
$\frac{5+7i}{3+i}\times \frac{3–i}{3–i}$
$=\frac{(5+7i)(3–i)}{(3+i)(3–i)}$

Step 2: Use the distributive property (FOIL):
$(5+7i)(3–i)$
$=5(3)+5(-i)+7i(3)+7i(-i)$

Step 3: Simplify each term:
$5(3)=15$
$5(-i)=-5i$
$7i(3)=21i$
$7i(-i)=-7i^2$

Step 4: Recall that $i^2=-1$:
$-7i^2$
$=7$

Step 5: Combine the terms in the numerator:
$15–5i+21i+7$
$=22+16i$

Step 6: Use the difference of squares formula in the denominator:
$(3+i)(3–i)=3^2–i^2$
$=9–(-1)=10$

Step 7: Write the final quotient:
$\frac{22+16i}{10}=\frac{22}{10}+\frac{16i}{10}$
$=2.2+1.6i$

Thus, the quotient is:
$z_1÷z_2=2.2+1.6i$

---

Example 47: Dividing Complex Numbers

Given two complex numbers $z_1=4+6i$ and $z_2=2–3i$, find their quotient.

Solution:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator:
$\frac{4+6i}{2–3i}\times \frac{2+3i}{2+3i}$
$=\frac{(4+6i)(2+3i)}{(2–3i)(2+3i)}$

Step 2: Use the distributive property (FOIL):
$(4+6i)(2+3i)$
$=4(2)+4(3i)+6i(2)+6i(3i)$

Step 3: Simplify each term:
$4(2)=8$
$4(3i)=12i$
$6i(2)=12i$
$6i(3i)=18i^2$

Step 4: Recall that $i^2=-1$:
$18i^2$
$=-18$

Step 5: Combine the terms in the numerator:
$8+12i+12i–18$
$=-10+24i$

Step 6: Use the difference of squares formula in the denominator:
$(2–3i)(2+3i)=2^2–(3i{)}^2$
$=4–(-9)$
$=13$

Step 7: Write the final quotient:
$\frac{-10+24i}{13}$
$=\frac{-10}{13}+\frac{24i}{13}$

Thus, the quotient is:
$z_1÷z_2=\frac{-10}{13}+\frac{24i}{13}$

---

Example 48: Multiplying Complex Numbers

Given two complex numbers $z_1=3+2i$ and $z_2=5–4i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(3+2i)(5–4i)$
$=3(5)+3(-4i)+2i(5)+2i(-4i)$

Step 2: Simplify each term:
$3(5)=15$
$3(-4i)=-12i$
$2i(5)=10i$
$2i(-4i)=-8i^2$

Step 3: Recall that $i^2=-1$:
$-8i^2$
$=8$

Step 4: Combine all terms:
$15–12i+10i+8$
$=23–2i$

Thus, the product is:
$z_1\times z_2=23–2i$

---

Example 49: Finding the Modulus of a Complex Number

Given a complex number $z=7–24i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=7$
$b=-24$

Step 3: Square the real part:
$7^2=49$

Step 4: Square the imaginary part:
$(-24{)}^2=576$

Step 5: Add the squares:
$49+576=625$

Step 6: Take the square root:
$\sqrt{625}=25$

Thus, the modulus is:
$|z|=25$

---

Example 50: Adding Complex Numbers

Given two complex numbers $z_1=8+3i$ and $z_2=6+7i$, find their sum.

Solution:

Step 1: Add the real parts:
$8+6=14$

Step 2: Add the imaginary parts:
$3i+7i=10i$

Thus, the sum is:
$z_1+z_2=14+10i$

---

Example 51: Subtracting Complex Numbers

Given two complex numbers $z_1=10+5i$ and $z_2=4+3i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$10–4=6$

Step 2: Subtract the imaginary parts:
$5i–3i=2i$

Thus, the difference is:
$z_1–z_2=6+2i$

---

Example 52: Dividing Complex Numbers

Given two complex numbers $z_1=9+6i$ and $z_2=3+2i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{9+6i}{3+2i}\times \frac{3–2i}{3–2i}$
$=\frac{(9+6i)(3–2i)}{(3+2i)(3–2i)}$

Step 2: Use the distributive property (FOIL):
$(9+6i)(3–2i)$
$=9(3)+9(-2i)+6i(3)+6i(-2i)$

Step 3: Simplify each term:
$9(3)=27$
$9(-2i)=-18i$
$6i(3)=18i$
$6i(-2i)=-12i^2$

Step 4: Recall that $i^2=-1$:
$-12i^2=12$

Step 5: Combine the terms in the numerator:
$27–18i+18i+12$
$=39$

Step 6: Use the difference of squares formula in the denominator:
$(3+2i)(3–2i)=3^2–(2i{)}^2$
$=9–(-4)$
$=13$

Step 7: Write the final quotient:
$\frac{39}{13}=3$

Thus, the quotient is:
$z_1÷z_2=3$

---

Example 53: Multiplying Complex Numbers

Given two complex numbers $z_1=6+2i$ and $z_2=3–i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(6+2i)(3–i)$
$=6(3)+6(-i)+2i(3)+2i(-i)$

Step 2: Simplify each term:
$6(3)=18$
$6(-i)=-6i$
$2i(3)=6i$
$2i(-i)=-2i^2$

Step 3: Recall that $i^2=-1$:
$-2i^2=2$

Step 4: Combine like terms:
$18–6i+6i+2$
$=20$

Thus, the product is:
$z_1\times z_2=20$

---

Example 54: Dividing Complex Numbers

Given two complex numbers $z_1=7+5i$ and $z_2=4–2i$, find their quotient.

Solution:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator:
$\frac{7+5i}{4–2i}\times \frac{4+2i}{4+2i}$
$=\frac{(7+5i)(4+2i)}{(4–2i)(4+2i)}$

Step 2: Use the distributive property (FOIL):
$(7+5i)(4+2i)$
$=7(4)+7(2i)+5i(4)+5i(2i)$

Step 3: Simplify each term:
$7(4)=28$
$7(2i)=14i$
$5i(4)=20i$
$5i(2i)=10i^2$

Step 4: Recall that $i^2=-1$:
$10i^2=-10$

Step 5: Combine the terms in the numerator:
$28+14i+20i–10$
$=18+34i$

Step 6: Use the difference of squares formula in the denominator:
$(4–2i)(4+2i)=4^2–(2i{)}^2$
$=16–(-4)$
$=20$

Step 7: Write the final quotient:
$\frac{18+34i}{20}=\frac{18}{20}+\frac{34i}{20}$
$=0.9+1.7i$

Thus, the quotient is:
$z_1÷z_2=0.9+1.7i$

---

Example 55: Finding the Modulus of a Complex Number

Given a complex number $z=5+12i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=5$
$b=12$

Step 3: Square the real part:
$5^2=25$

Step 4: Square the imaginary part:
${12}^2=144$

Step 5: Add the squares:
$25+144=169$

Step 6: Take the square root:
$\sqrt{169}=13$

Thus, the modulus is:
$|z|=13$

---

Example 56: Adding Complex Numbers

Given two complex numbers $z_1=9+4i$ and $z_2=3+7i$, find their sum.

Solution:

Step 1: Add the real parts:
$9+3=12$

Step 2: Add the imaginary parts:
$4i+7i=11i$

Thus, the sum is:
$z_1+z_2=12+11i$

---

Example 57: Subtracting Complex Numbers

Given two complex numbers $z_1=8+6i$ and $z_2=5+2i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$8–5=3$

Step 2: Subtract the imaginary parts:
$6i–2i=4i$

Thus, the difference is:
$z_1–z_2=3+4i$

---

Example 58: Dividing Complex Numbers

Given two complex numbers $z_1=10+4i$ and $z_2=2+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{10+4i}{2+i}\times \frac{2–i}{2–i}$
$=\frac{(10+4i)(2–i)}{(2+i)(2–i)}$

Step 2: Use the distributive property (FOIL):
$(10+4i)(2–i)$
$=10(2)+10(-i)+4i(2)+4i(-i)$

Step 3: Simplify each term:
$10(2)=20$
$10(-i)=-10i$
$4i(2)=8i$
$4i(-i)=-4i^2$

Step 4: Recall that $i^2=-1$:
$-4i^2=4$

Step 5: Combine the terms in the numerator:
$20–10i+8i+4$
$=24–2i$

Step 6: Use the difference of squares formula in the denominator:
$(2+i)(2–i)=2^2–i^2$
$=4–(-1)$
$=5$

Step 7: Write the final quotient:
$\frac{24–2i}{5}=\frac{24}{5}–\frac{2i}{5}$

Thus, the quotient is:
$z_1÷z_2=\frac{24}{5}–\frac{2i}{5}$

---

Example 59: Multiplying Complex Numbers

Given two complex numbers $z_1=3+7i$ and $z_2=2–6i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(3+7i)(2–6i)$
$=3(2)+3(-6i)+7i(2)+7i(-6i)$

Step 2: Simplify each term:
$3(2)=6$
$3(-6i)=-18i$
$7i(2)=14i$
$7i(-6i)=-42i^2$

Step 3: Recall that $i^2=-1$:
$-42i^2$
$=42$

Step 4: Combine like terms:
$6–18i+14i+42$
$=48–4i$

Thus, the product is:
$z_1\times z_2=48–4i$

---

Example 60: Dividing Complex Numbers

Given two complex numbers $z_1=8+3i$ and $z_2=4–i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{8+3i}{4–i}\times \frac{4+i}{4+i}$
$=\frac{(8+3i)(4+i)}{(4–i)(4+i)}$

Step 2: Use the distributive property (FOIL):
$(8+3i)(4+i)$
$=8(4)+8(i)+3i(4)+3i(i)$

Step 3: Simplify each term:
$8(4)=32$
$8(i)=8i$
$3i(4)=12i$
$3i(i)=3i^2$

Step 4: Recall that $i^2=-1$:
$3i^2=-3$

Step 5: Combine the terms in the numerator:
$32+8i+12i–3$
$=29+20i$

Step 6: Use the difference of squares formula in the denominator:
$(4–i)(4+i)=4^2–i^2$
$=16–(-1)$
$=17$

Step 7: Write the final quotient:
$\frac{29+20i}{17}=\frac{29}{17}+\frac{20i}{17}$

Thus, the quotient is:
$z_1÷z_2=\frac{29}{17}+\frac{20i}{17}$

---

Example 61: Finding the Modulus of a Complex Number

Given a complex number $z=5+8i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=5$
$b=8$

Step 3: Square the real part:
$5^2=25$

Step 4: Square the imaginary part:
$8^2=64$

Step 5: Add the squares:
$25+64=89$

Step 6: Take the square root:
$\sqrt{89}$

Thus, the modulus is:
$|z|=\sqrt{89}$

---

Example 62: Adding Complex Numbers

Given two complex numbers $z_1=10+4i$ and $z_2=7+2i$, find their sum.

Solution:

Step 1: Add the real parts:
$10+7=17$

Step 2: Add the imaginary parts:
$4i+2i=6i$

Thus, the sum is:
$z_1+z_2=17+6i$

---

Example 63: Subtracting Complex Numbers

Given two complex numbers $z_1=12+5i$ and $z_2=8+3i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$12–8=4$

Step 2: Subtract the imaginary parts:
$5i–3i=2i$

Thus, the difference is:
$z_1–z_2=4+2i$

---

Example 64: Dividing Complex Numbers

Given two complex numbers $z_1=9+7i$ and $z_2=5+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{9+7i}{5+i}\times \frac{5–i}{5–i}$
$=\frac{(9+7i)(5–i)}{(5+i)(5–i)}$

Step 2: Use the distributive property (FOIL):
$(9+7i)(5–i)$
$=9(5)+9(-i)+7i(5)+7i(-i)$

Step 3: Simplify each term:
$9(5)=45$
$9(-i)=-9i$
$7i(5)=35i$
$7i(-i)=-7i^2$

Step 4: Recall that $i^2=-1$:
$-7i^2=7$

Step 5: Combine the terms in the numerator:
$45–9i+35i+7$
$=52+26i$

Step 6: Use the difference of squares formula in the denominator:
$(5+i)(5–i)=5^2–i^2$
$=25–(-1)=26$

Step 7: Write the final quotient:
$\frac{52+26i}{26}=\frac{52}{26}+\frac{26i}{26}$
$=2+i$

Thus, the quotient is:
$z_1÷z_2=2+i$

---

Example 65: Multiplying Complex Numbers

Given two complex numbers $z_1=2+3i$ and $z_2=1–4i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(2+3i)(1–4i)$
$=2(1)+2(-4i)+3i(1)+3i(-4i)$

Step 2: Simplify each term:
$2(1)=2$
$2(-4i)=-8i$
$3i(1)=3i$
$3i(-4i)=-12i^2$

Step 3: Recall that $i^2=-1$:
$-12i^2=12$

Step 4: Combine like terms:
$2–8i+3i+12$
$=14–5i$

Thus, the product is:
$z_1\times z_2=14–5i$

---

Example 66: Dividing Complex Numbers

Given two complex numbers $z_1=6+8i$ and $z_2=4+2i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{6+8i}{4+2i}\times \frac{4–2i}{4–2i}$
$=\frac{(6+8i)(4–2i)}{(4+2i)(4–2i)}$

Step 2: Use the distributive property (FOIL):
$(6+8i)(4–2i)$
$=6(4)+6(-2i)+8i(4)+8i(-2i)$

Step 3: Simplify each term:
$6(4)=24$
$6(-2i)=-12i$
$8i(4)=32i$
$8i(-2i)=-16i^2$

Step 4: Recall that $i^2=-1$:
$-16i^2=16$

Step 5: Combine the terms in the numerator:
$24–12i+32i+16$
$=40+20i$

Step 6: Use the difference of squares formula in the denominator:
$(4+2i)(4–2i)=4^2–(2i{)}^2$
$=16–(-4)$
$=20$

Step 7: Write the final quotient:
$\frac{40+20i}{20}=\frac{40}{20}+\frac{20i}{20}$
$=2+i$

Thus, the quotient is:
$z_1÷z_2=2+i$

---

Example 67: Finding the Modulus of a Complex Number

Given a complex number $z=9+12i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=9$
$b=12$

Step 3: Square the real part:
$9^2=81$

Step 4: Square the imaginary part:
${12}^2=144$

Step 5: Add the squares:
$81+144=225$

Step 6: Take the square root:
$\sqrt{225}=15$

Thus, the modulus is:
$|z|=15$

---

Example 68: Adding Complex Numbers

Given two complex numbers $z_1=7+9i$ and $z_2=3+5i$, find their sum.

Solution:

Step 1: Add the real parts:
$7+3=10$

Step 2: Add the imaginary parts:
$9i+5i=14i$

Thus, the sum is:
$z_1+z_2=10+14i$

---

Example 69: Subtracting Complex Numbers

Given two complex numbers $z_1=10+8i$ and $z_2=6+3i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$10–6=4$

Step 2: Subtract the imaginary parts:
$8i–3i=5i$

Thus, the difference is:
$z_1–z_2=4+5i$

---

Example 70: Dividing Complex Numbers

Given two complex numbers $z_1=11+4i$ and $z_2=5–i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{11+4i}{5–i}\times \frac{5+i}{5+i}$
$=\frac{(11+4i)(5+i)}{(5–i)(5+i)}$

Step 2: Use the distributive property (FOIL):
$(11+4i)(5+i)$
$=11(5)+11(i)+4i(5)+4i(i)$

Step 3: Simplify each term:
$11(5)=55$
$11(i)=11i$
$4i(5)=20i$
$4i(i)=4i^2$

Step 4: Recall that $i^2=-1$:
$4i^2=-4$

Step 5: Combine the terms in the numerator:
$55+11i+20i–4$
$=51+31i$

Step 6: Use the difference of squares formula in the denominator:
$(5–i)(5+i)=5^2–i^2$
$=25–(-1)$
$=26$

Step 7: Write the final quotient:
$\frac{51+31i}{26}=\frac{51}{26}+\frac{31i}{26}$

Thus, the quotient is:
$z_1÷z_2=\frac{51}{26}+\frac{31i}{26}$

---

Example 71: Multiplying Complex Numbers

Given two complex numbers $z_1=5+6i$ and $z_2=2–3i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(5+6i)(2–3i)$
$=5(2)+5(-3i)+6i(2)+6i(-3i)$

Step 2: Simplify each term:
$5(2)=10$
$5(-3i)=-15i$
$6i(2)=12i$
$6i(-3i)=-18i^2$

Step 3: Recall that $i^2=-1$:
$-18i^2=18$

Step 4: Combine like terms:
$10–15i+12i+18$
$=28–3i$

Thus, the product is:
$z_1\times z_2=28–3i$

---

Example 72: Dividing Complex Numbers

Given two complex numbers $z_1=8+4i$ and $z_2=3+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{8+4i}{3+i}\times \frac{3–i}{3–i}$
$=\frac{(8+4i)(3–i)}{(3+i)(3–i)}$

Step 2: Use the distributive property (FOIL):
$(8+4i)(3–i)$
$=8(3)+8(-i)+4i(3)+4i(-i)$

Step 3: Simplify each term:
$8(3)=24$
$8(-i)=-8i$
$4i(3)=12i$
$4i(-i)=-4i^2$

Step 4: Recall that $i^2=-1$:
$-4i^2=4$

Step 5: Combine the terms in the numerator:
$24–8i+12i+4$
$=28+4i$

Step 6: Use the difference of squares formula in the denominator:
$(3+i)(3–i)=3^2–i^2$
$=9–(-1)=10$

Step 7: Write the final quotient:
$\frac{28+4i}{10}=\frac{28}{10}+\frac{4i}{10}$
$=2.8+0.4i$

Thus, the quotient is:
$z_1÷z_2=2.8+0.4i$

---

Example 73: Finding the Modulus of a Complex Number

Given a complex number $z=7+24i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=7$
$b=24$

Step 3: Square the real part:
$7^2=49$

Step 4: Square the imaginary part:
${24}^2=576$

Step 5: Add the squares:
$49+576=625$

Step 6: Take the square root:
$\sqrt{625}=25$

Thus, the modulus is:
$|z|=25$

---

Example 74: Adding Complex Numbers

Given two complex numbers $z_1=9+5i$ and $z_2=6+2i$, find their sum.

Solution:

Step 1: Add the real parts:
$9+6=15$

Step 2: Add the imaginary parts:
$5i+2i=7i$

Thus, the sum is:
$z_1+z_2=15+7i$

---

Example 75: Subtracting Complex Numbers

Given two complex numbers $z_1=10+8i$ and $z_2=4+3i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$10–4=6$

Step 2: Subtract the imaginary parts:
$8i–3i=5i$

Thus, the difference is:
$z_1–z_2=6+5i$

---

Example 76: Multiplying Complex Numbers

Given two complex numbers $z_1=5+7i$ and $z_2=3–2i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(5+7i)(3–2i)$
$=5(3)+5(-2i)+7i(3)+7i(-2i)$

Step 2: Simplify each term:
$5(3)=15$
$5(-2i)=-10i$
$7i(3)=21i$
$7i(-2i)=-14i^2$

Step 3: Recall that $i^2=-1$:
$-14i^2$
$=14$

Step 4: Combine like terms:
$15–10i+21i+14$
$=29+11i$

Thus, the product is:
$z_1\times z_2=29+11i$

---

Example 77: Dividing Complex Numbers

Given two complex numbers $z_1=7+5i$ and $z_2=3+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator:
$\frac{7+5i}{3+i}\times \frac{3–i}{3–i}$
$=\frac{(7+5i)(3–i)}{(3+i)(3–i)}$

Step 2: Use the distributive property (FOIL):
$(7+5i)(3–i)$
$=7(3)+7(-i)+5i(3)+5i(-i)$

Step 3: Simplify each term:
$7(3)=21$
$7(-i)=-7i$
$5i(3)=15i$
$5i(-i)=-5i^2$

Step 4: Recall that $i^2=-1$:
$-5i^2=5$

Step 5: Combine the terms in the numerator:
$21–7i+15i+5$
$=26+8i$

Step 6: Use the difference of squares formula in the denominator:
$(3+i)(3–i)=3^2–i^2$
$=9–(-1)=10$

Step 7: Write the final quotient:
$\frac{26+8i}{10}=\frac{26}{10}+\frac{8i}{10}$
$=2.6+0.8i$

Thus, the quotient is:
$z_1÷z_2=2.6+0.8i$

---

Example 78: Finding the Modulus of a Complex Number

Given a complex number $z=4+3i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=4$
$b=3$

Step 3: Square the real part:
$4^2=16$

Step 4: Square the imaginary part:
$3^2=9$

Step 5: Add the squares:
$16+9=25$

Step 6: Take the square root:
$\sqrt{25}=5$

Thus, the modulus is:
$|z|=5$

---

Example 79: Adding Complex Numbers

Given two complex numbers $z_1=6+4i$ and $z_2=2+3i$, find their sum.

Solution:

Step 1: Add the real parts:
$6+2=8$

Step 2: Add the imaginary parts:
$4i+3i=7i$

Thus, the sum is:
$z_1+z_2=8+7i$

---

Example 80: Subtracting Complex Numbers

Given two complex numbers $z_1=9+7i$ and $z_2=4+2i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$9–4=5$

Step 2: Subtract the imaginary parts:
$7i–2i=5i$

Thus, the difference is:
$z_1–z_2=5+5i$

---

Example 81: Multiplying Complex Numbers

Given two complex numbers $z_1=3+4i$ and $z_2=1–2i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(3+4i)(1–2i)$
$=3(1)+3(-2i)+4i(1)+4i(-2i)$

Step 2: Simplify each term:
$3(1)=3$
$3(-2i)=-6i$
$4i(1)=4i$
$4i(-2i)=-8i^2$

Step 3: Recall that $i^2=-1$:
$-8i^2=8$

Step 4: Combine like terms:
$3–6i+4i+8$
$=11–2i$

Thus, the product is:
$z_1\times z_2=11–2i$

---

Example 82: Dividing Complex Numbers

Given two complex numbers $z_1=6+5i$ and $z_2=2+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{6+5i}{2+i}\times \frac{2–i}{2–i}$
$=\frac{(6+5i)(2–i)}{(2+i)(2–i)}$

Step 2: Use the distributive property (FOIL):
$(6+5i)(2–i)$
$=6(2)+6(-i)+5i(2)+5i(-i)$

Step 3: Simplify each term:
$6(2)=12$
$6(-i)=-6i$
$5i(2)=10i$
$5i(-i)=-5i^2$

Step 4: Recall that $i^2=-1$:
$-5i^2=5$

Step 5: Combine the terms in the numerator:
$12–6i+10i+5$
$=17+4i$

Step 6: Use the difference of squares formula in the denominator:
$(2+i)(2–i)=2^2–i^2$
$=4–(-1)=5$

Step 7: Write the final quotient:
$\frac{17+4i}{5}=\frac{17}{5}+\frac{4i}{5}$

Thus, the quotient is:
$z_1÷z_2=\frac{17}{5}+\frac{4i}{5}$

---

Example 83: Finding the Modulus of a Complex Number

Given a complex number $z=4+6i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=4$
$b=6$

Step 3: Square the real part:
$4^2=16$

Step 4: Square the imaginary part:
$6^2=36$

Step 5: Add the squares:
$16+36=52$

Step 6: Take the square root:
$\sqrt{52}$

Thus, the modulus is:
$|z|=\sqrt{52}$

---

Example 84: Adding Complex Numbers

Given two complex numbers $z_1=7+8i$ and $z_2=4+3i$, find their sum.

Solution:

Step 1: Add the real parts:
$7+4=11$

Step 2: Add the imaginary parts:
$8i+3i=11i$

Thus, the sum is:
$z_1+z_2=11+11i$

---

Example 85: Subtracting Complex Numbers

Given two complex numbers $z_1=10+6i$ and $z_2=3+2i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$10–3=7$

Step 2: Subtract the imaginary parts:
$6i–2i=4i$

Thus, the difference is:
$z_1–z_2=7+4i$

---

Example 86: Multiplying Complex Numbers

Given two complex numbers $z_1=2+7i$ and $z_2=4+3i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(2+7i)(4+3i)$
$=2(4)+2(3i)+7i(4)+7i(3i)$

Step 2: Simplify each term:
$2(4)=8$
$2(3i)=6i$
$7i(4)=28i$
$7i(3i)=21i^2$

Step 3: Recall that $i^2=-1$:
$21i^2=-21$

Step 4: Combine like terms:
$8+6i+28i–21$
$=-13+34i$

Thus, the product is:
$z_1\times z_2=-13+34i$

---

Example 87: Dividing Complex Numbers

Given two complex numbers $z_1=9+6i$ and $z_2=5+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{9+6i}{5+i}\times \frac{5–i}{5–i}$
$=\frac{(9+6i)(5–i)}{(5+i)(5–i)}$

Step 2: Use the distributive property (FOIL):
$(9+6i)(5–i)$
$=9(5)+9(-i)+6i(5)+6i(-i)$

Step 3: Simplify each term:
$9(5)=45$
$9(-i)=-9i$
$6i(5)=30i$
$6i(-i)=-6i^2$

Step 4: Recall that $i^2=-1$:
$-6i^2=6$

Step 5: Combine the terms in the numerator:
$45–9i+30i+6$
$=51+21i$

Step 6: Use the difference of squares formula in the denominator:
$(5+i)(5–i)=5^2–i^2$
$=25–(-1)=26$

Step 7: Write the final quotient:
$\frac{51+21i}{26}=\frac{51}{26}+\frac{21i}{26}$

Thus, the quotient is:
$z_1÷z_2=\frac{51}{26}+\frac{21i}{26}$

---

Example 88: Multiplying Complex Numbers

Given two complex numbers $z_1=3+2i$ and $z_2=5–i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(3+2i)(5–i)$
$=3(5)+3(-i)+2i(5)+2i(-i)$

Step 2: Simplify each term:
$3(5)=15$
$3(-i)=-3i$
$2i(5)=10i$
$2i(-i)=-2i^2$

Step 3: Recall that $i^2=-1$:
$-2i^2=2$

Step 4: Combine like terms:
$15–3i+10i+2$
$=17+7i$

Thus, the product is:
$z_1\times z_2=17+7i$

---

Example 89: Dividing Complex Numbers

Given two complex numbers $z_1=8+4i$ and $z_2=3+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{8+4i}{3+i}\times \frac{3–i}{3–i}$
$=\frac{(8+4i)(3–i)}{(3+i)(3–i)}$

Step 2: Use the distributive property (FOIL):
$(8+4i)(3–i)$
$=8(3)+8(-i)+4i(3)+4i(-i)$

Step 3: Simplify each term:
$8(3)=24$
$8(-i)=-8i$
$4i(3)=12i$
$4i(-i)=-4i^2$

Step 4: Recall that $i^2=-1$:
$-4i^2=4$

Step 5: Combine the terms in the numerator:
$24–8i+12i+4$
$=28+4i$

Step 6: Use the difference of squares formula in the denominator:
$(3+i)(3–i)=3^2–i^2$
$=9–(-1)=10$

Step 7: Write the final quotient:
$\frac{28+4i}{10}=\frac{28}{10}+\frac{4i}{10}$
$=2.8+0.4i$

Thus, the quotient is:
$z_1÷z_2=2.8+0.4i$

---

Example 90: Finding the Modulus of a Complex Number

Given a complex number $z=6+8i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=6$
$b=8$

Step 3: Square the real part:
$6^2=36$

Step 4: Square the imaginary part:
$8^2=64$

Step 5: Add the squares:
$36+64=100$

Step 6: Take the square root:
$\sqrt{100}=10$

Thus, the modulus is:
$|z|=10$

---

Example 91: Adding Complex Numbers

Given two complex numbers $z_1=9+5i$ and $z_2=3+4i$, find their sum.

Solution:

Step 1: Add the real parts:
$9+3=12$

Step 2: Add the imaginary parts:
$5i+4i=9i$

Thus, the sum is:
$z_1+z_2=12+9i$

---

Example 92: Subtracting Complex Numbers

Given two complex numbers $z_1=7+6i$ and $z_2=2+3i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$7–2=5$

Step 2: Subtract the imaginary parts:
$6i–3i=3i$

Thus, the difference is:
$z_1–z_2=5+3i$

---

Example 93: Dividing Complex Numbers

Given two complex numbers $z_1=11+3i$ and $z_2=5+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{11+3i}{5+i}\times \frac{5–i}{5–i}$
$=\frac{(11+3i)(5–i)}{(5+i)(5–i)}$

Step 2: Use the distributive property (FOIL):
$(11+3i)(5–i)$
$=11(5)+11(-i)+3i(5)+3i(-i)$

Step 3: Simplify each term:
$11(5)=55$
$11(-i)=-11i$
$3i(5)=15i$
$3i(-i)=-3i^2$

Step 4: Recall that $i^2=-1$:
$-3i^2=3$

Step 5: Combine the terms in the numerator:
$55–11i+15i+3$
$=58+4i$

Step 6: Use the difference of squares formula in the denominator:
$(5+i)(5–i)=5^2–i^2$
$=25–(-1)=26$

Step 7: Write the final quotient:
$\frac{58+4i}{26}=\frac{58}{26}+\frac{4i}{26}$
$=2.23+0.15i$

Thus, the quotient is:
$z_1÷z_2=2.23+0.15i$

---

Example 94: Multiplying Complex Numbers

Given two complex numbers $z_1=2+3i$ and $z_2=5–4i$, find their product.

Solution:

Step 1: Use the distributive property (FOIL):
$(2+3i)(5–4i)$
$=2(5)+2(-4i)+3i(5)+3i(-4i)$

Step 2: Simplify each term:
$2(5)=10$
$2(-4i)=-8i$
$3i(5)=15i$
$3i(-4i)=-12i^2$

Step 3: Recall that $i^2=-1$:
$-12i^2=12$

Step 4: Combine like terms:
$10–8i+15i+12$
$=22+7i$

Thus, the product is:
$z_1\times z_2=22+7i$

---

Example 95: Dividing Complex Numbers

Given two complex numbers $z_1=7+2i$ and $z_2=4+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{7+2i}{4+i}\times \frac{4–i}{4–i}$
$=\frac{(7+2i)(4–i)}{(4+i)(4–i)}$

Step 2: Use the distributive property (FOIL):
$(7+2i)(4–i)$
$=7(4)+7(-i)+2i(4)+2i(-i)$

Step 3: Simplify each term:
$7(4)=28$
$7(-i)=-7i$
$2i(4)=8i$
$2i(-i)=-2i^2$

Step 4: Recall that $i^2=-1$:
$-2i^2=2$

Step 5: Combine the terms in the numerator:
$28–7i+8i+2$
$=30+i$

Step 6: Use the difference of squares formula in the denominator:
$(4+i)(4–i)=4^2–i^2$
$=16–(-1)=17$

Step 7: Write the final quotient:
$\frac{30+i}{17}=\frac{30}{17}+\frac{i}{17}$

Thus, the quotient is:
$z_1÷z_2=\frac{30}{17}+\frac{i}{17}$

---

Example 96: Finding the Modulus of a Complex Number

Given a complex number $z=3+4i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=3$
$b=4$

Step 3: Square the real part:
$3^2=9$

Step 4: Square the imaginary part:
$4^2=16$

Step 5: Add the squares:
$9+16=25$

Step 6: Take the square root:
$\sqrt{25}=5$

Thus, the modulus is:
$|z|=5$

---

Example 97: Adding Complex Numbers

Given two complex numbers $z_1=6+5i$ and $z_2=4+3i$, find their sum.

Solution:

Step 1: Add the real parts:
$6+4=10$

Step 2: Add the imaginary parts:
$5i+3i=8i$

Thus, the sum is:
$z_1+z_2=10+8i$

---

Example 98: Subtracting Complex Numbers

Given two complex numbers $z_1=9+7i$ and $z_2=3+2i$, find their difference.

Solution:

Step 1: Subtract the real parts:
$9–3=6$

Step 2: Subtract the imaginary parts:
$7i–2i=5i$

Thus, the difference is:
$z_1–z_2=6+5i$

---

Example 99: Dividing Complex Numbers

Given two complex numbers $z_1=5+2i$ and $z_2=1+i$, find their quotient.

Solution:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator:
$\frac{5+2i}{1+i}\times \frac{1–i}{1–i}$
$=\frac{(5+2i)(1–i)}{(1+i)(1–i)}$

Step 2: Use the distributive property (FOIL):
$(5+2i)(1–i)$
$=5(1)+5(-i)+2i(1)+2i(-i)$

Step 3: Simplify each term:
$5(1)=5$
$5(-i)=-5i$
$2i(1)=2i$
$2i(-i)=-2i^2$

Step 4: Recall that $i^2=-1$:
$-2i^2=2$

Step 5: Combine the terms in the numerator:
$5–5i+2i+2$
$=7–3i$

Step 6: Use the difference of squares formula in the denominator:
$(1+i)(1–i)=1^2–i^2$
$=1–(-1)=2$

Step 7: Write the final quotient:
$\frac{7–3i}{2}=\frac{7}{2}–\frac{3i}{2}$

Thus, the quotient is:
$z_1÷z_2=\frac{7}{2}–\frac{3i}{2}$

---

Example 100: Finding the Modulus of a Complex Number

Given a complex number $z=12+5i$, find its modulus.

Solution:

Step 1: Use the modulus formula:
$|z|=\sqrt{a^2+b^2}$

Step 2: Identify the real and imaginary parts:
$a=12$
$b=5$

Step 3: Square the real part:
${12}^2=144$

Step 4: Square the imaginary part:
$5^2=25$

Step 5: Add the squares:
$144+25=169$

Step 6: Take the square root:
$\sqrt{169}=13$

Thus, the modulus is:
$|z|=13$