This stack focuses on the fundamental operations of addition and subtraction of complex numbers. The core principle is combining like terms: adding/subtracting real parts with real parts and imaginary parts with imaginary parts.
What is the rule for adding two complex numbers, \((a + bi)\) and \((c + di)\)?
To add complex numbers, add the real parts together and add the imaginary parts together. The formula is: \\$$ (a + c) + (b + d)i $$
Calculate the sum: \((5 + 2i) + (3 + 6i)\).
Add the real parts: \(5 + 3 = 8\). Add the imaginary parts: \(2i + 6i = 8i\). The result is \(8 + 8i\).
Calculate the sum: \((7 - 4i) + (-2 + 9i)\).
Real parts: \(7 + (-2) = 5\). Imaginary parts: \(-4i + 9i = 5i\). The result is \(5 + 5i\).
What is the rule for subtracting one complex number from another, \((a + bi) - (c + di)\)?
To subtract complex numbers, subtract the real parts and subtract the imaginary parts. Distribute the negative sign first. The formula is: \\$$ (a - c) + (b - d)i $$
Calculate the difference: \((9 + 5i) - (4 + 2i)\).
Subtract the real parts: \(9 - 4 = 5\). Subtract the imaginary parts: \(5i - 2i = 3i\). The result is \(5 + 3i\).
Calculate the difference: \((3 - 8i) - (7 - 2i)\).
Rewrite as \(3 - 8i - 7 + 2i\). Real parts: \(3 - 7 = -4\). Imaginary parts: \(-8i + 2i = -6i\). The result is \(-4 - 6i\).
What is the additive identity for complex numbers?
The additive identity is \(0\), which can be written in standard form as \(0 + 0i\). Adding \(0 + 0i\) to any complex number does not change its value.
What is the additive inverse of the complex number \(z = a + bi\)?
The additive inverse is \(-z = -a - bi\). When a complex number is added to its additive inverse, the result is the additive identity, \(0 + 0i\).
Find the additive inverse of \(-4 + 11i\).
The additive inverse is \(-(-4 + 11i)\), which simplifies to \(4 - 11i\).
