Input two complex numbers, select an operation (addition, subtraction, multiplication, or division), and instantly calculate the result.
A complex number is composed of a real part and an imaginary part, expressed as \( a + bi \), where: \( a \) is the real part, \( b \) is the imaginary part and \( i \) is the imaginary unit (\( i^2 = -1 \)).
To add two complex numbers, add the real parts and the imaginary parts separately. \( (a + bi) + (c + di) = (a + c) + (b + d)i \)
To subtract two complex numbers, subtract the real parts and the imaginary parts separately. \( (a + bi) - (c + di) = (a - c) + (b - d)i \)
To multiply two complex numbers, apply the distributive property and simplify. \( (a + bi) \times (c + di) = ac + adi + bci + bdi^2 \) Since \(i^2 = -1\), the result is: \( (a + bi) \times (c + di) = (ac - bd) + (ad + bc)i \)
To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For \( (a + bi) \div (c + di) \): \( \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} \) The result simplifies to: \( \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \)
Solution:
\( (3 + 2i) + (1 + 4i) \)
\( = (3 + 1) + (2 + 4)i \)
\( = 4 + 6i \)
Result: \( 4 + 6i \)
Solution:
\( (2 + 3i) \times (1 - i) \)
\( = 2(1) + 2(-i) + 3i(1) + 3i(-i) \)
\( = 2 - 2i + 3i - 3i^2 \)
\( = 2 - 2i + 3i + 3 \)
\( = (2 + 3) + (-2 + 3)i \)
\( = 5 + i \)
Result: \( 5 + i \)
Solution:
\( \frac{4 + 2i}{1 + i} \times \frac{1 - i}{1 - i} = \frac{(4 + 2i)(1 - i)}{(1 + i)(1 - i)} = \frac{(4 + 2i)(1 - i)}{2} \)
Simplify:
\( (4 + 2i)(1 - i) \)
\( = 4(1) + 4(-i) + 2i(1) + 2i(-i) \)
\( = 4 - 4i + 2i - 2i^2 \)
\( = 4 - 2i + 2 \)
\( = 6 - 2i \)
So:
\( \frac{(4 + 2i)(1 - i)}{2} = \frac{6 - 2i}{2} = 3 - i \)
Result: \( 3 - i \)