Question

Z1=1+4i; z2=4-i вычислить z1/z2

Answer 1

z1/z2= (1+4i)/(4 -i) =(1+4i)(4+i)/( 16+1)=(4+17i+4i²)/17=17i/17= i.

Answer 2

To calculate $\frac{z1}{z2}$, you need to perform complex division. The formula for dividing two complex numbers is as follows:

$\frac{z1}{z2} = \frac{{\text{Re}(z1) + i \cdot \text{Im}(z1)}}{{\text{Re}(z2) + i \cdot \text{Im}(z2)}}$

where $\text{Re}(z)$ is the real part of complex number $z$, and $\text{Im}(z)$ is the imaginary part of complex number $z$.

Given $z1 = 1 + 4i$ and $z2 = 4 - i$, let's compute $\frac{z1}{z2}$:

Step 1: Find the real and imaginary parts of $z1$ and $z2$: $\text{Re}(z1) = 1 \quad \text{Im}(z1) = 4$ $\text{Re}(z2) = 4 \quad \text{Im}(z2) = -1$

Step 2: Apply the complex division formula: $\frac{z1}{z2} = \frac{{1 + 4i}}{{4 - i}}$

Step 3: Rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator (i.e., changing the sign of the imaginary part): $\frac{z1}{z2} = \frac{{(1 + 4i) \cdot (4 + i)}}{{(4 - i) \cdot (4 + i)}}$

Step 4: Simplify the expression: $\frac{z1}{z2} = \frac{{4 + 17i + 4i^2}}{{16 + i^2}}$

Note that $i^2 = -1$, so we have: $\frac{z1}{z2} = \frac{{4 + 17i - 4}}{{16 - 1}} = \frac{{17 + 17i}}{{15}}$

Thus, $\frac{z1}{z2} = \frac{{17}}{{15}} + \frac{{17i}}{{15}}$.