The real part of the complex number 4 + 5i is 4.

|z| = √((-2)^2 + (2)^2) = √(4 + 4) = √8 = 2√2.

z^2 = (1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i.

z² = (1 + i)² = 1² + 2(1)(i) + i² = 1 + 2i - 1 = 2i.

|z|^2 = (2^2 + 2^2) = 4 + 4 = 8.

The conjugate of z = 2 - 3i is 2 + 3i.

The imaginary part of the complex number z = 3 + 4i is 4.

z + z* = (3 + 4i) + (3 - 4i) = 6.

z + z̅ = (3 + 4i) + (3 - 4i) = 6.

The real part of the complex number z = 4 + 3i is 4.

The argument of z = 5 + 12i is given by arctan(12/5).

The argument of -1 + 0i is π (180 degrees) since it lies on the negative x-axis.

The modulus is calculated as √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

(1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 2.

(2 + 3i)(4 - i) = 8 - 2i + 12i - 3i^2 = 8 + 10i + 3 = 11 + 10i.

The real part of the complex number 4 + 5i is 4.

The real part of the complex number 7 - 4i is 7.

The modulus is √(1^2 + 2^2) = √(1 + 4) = √5. The square of the modulus is 5.

The modulus is √(1^2 + 1^2) = √2, and the square of the modulus is 2.

(3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i.

(1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2.

Multiply numerator and denominator by the conjugate of the denominator: (2 + 3i)(1 - i) / (1 + i)(1 - i) = (2 - 2i + 3i + 3) / (1 + 1) = (5 + i) / 2 = 2.5 + 0.5i.

Using the difference of squares: (a + b)(a - b) = a² - b². Here, (2)² - (3i)² = 4 - (-9) = 4 + 9 = 13.

Multiplying numerator and denominator by the conjugate of the denominator: (3 + 4i)(1 - i) / (1 + i)(1 - i) = (3 - 3i + 4i + 4) / 2 = (7 + i) / 2 = 2 + i.

The conjugate of a complex number a + bi is a - bi. Thus, the conjugate of 2 - 3i is 2 + 3i.

The conjugate of a complex number a + bi is a - bi. Thus, the conjugate of 5 - 3i is 5 + 3i.