What is the result of multiplying two complex numbers, u1 and v1?

To understand the result of multiplying two complex numbers u1 and v1, we can break it down into its constituents. Suppose we have:

u1 = a + bi and v1 = c + di,

where:

• a and c are the real parts of u1 and v1, respectively,
• b and d are the imaginary parts of u1 and v1, respectively.

The multiplication of two complex numbers follows the distributive property, so we expand:

uv = (a + bi)(c + di)

Applying the distributive law, we get:

• ac (real part) + adi + bci + bdi²

Since i² = -1, the equation simplifies to:

• Real part: ac – bd
• Imaginary part: ad + bc

Therefore, the product of the two complex numbers u1 and v1 can be expressed as:

uv = (ac – bd) + (ad + bc)i

In summary, multiplying complex numbers involves both multiplying their real parts and imaginary parts and combining them appropriately. The result retains both the real and imaginary components, making complex numbers versatile in various mathematical applications.