Determine the real and imaginary parts of the complex…
Determine the real and imaginary parts of the complex numbers: $$ 1^{\circ}\; -7i; \quad 2^{\circ}\; 2-4i; \quad 3^{\circ}\; \cos\alpha + i\sin\alpha - 1; \quad 4^{\circ}\; a+ib+c+i. $$
Explanation of Method
Every complex number can be written in standard form $z = x + iy$, where $x = \text{Re}(z)$ is the real part and $y = \text{Im}(z)$ is the imaginary part. Both $x$ and $y$ are real numbers, and $i$ is the imaginary unit with $i^2 = -1$.
Important: the imaginary part is the coefficient of $i$, not $i$ itself.
- Group all terms without $i$ — these form the real part.
- Group all coefficients of $i$ — these form the imaginary part.
Step 1° · $z = -7i$
Writing in standard form:
$$ z = -7i = 0 + i \cdot (-7) $$
$$ \boxed{\text{Re}(z) = 0, \quad \text{Im}(z) = -7} $$
This is a purely imaginary number on the negative imaginary axis.
Step 2° · $z = 2 - 4i$
Writing in standard form:
$$ z = 2 - 4i = 2 + i \cdot (-4) $$
$$ \boxed{\text{Re}(z) = 2, \quad \text{Im}(z) = -4} $$
This complex number has both a nonzero real part and a nonzero imaginary part.
Step 3° · $z = \cos\alpha + i\sin\alpha - 1$
Group the real and imaginary terms:
$$ z = \cos\alpha + i\sin\alpha - 1 = (\cos\alpha - 1) + i\sin\alpha $$
$$ \boxed{\text{Re}(z) = \cos\alpha - 1, \quad \text{Im}(z) = \sin\alpha} $$
Note: both parts depend on the parameter $\alpha$.
Special cases:
- If $\alpha = 0$: $\text{Re}(z) = 1 - 1 = 0$, $\text{Im}(z) = 0$, so $z = 0$
- If $\alpha = \dfrac{\pi}{2}$: $\text{Re}(z) = 0 - 1 = -1$, $\text{Im}(z) = 1$, so $z = -1 + i$
- If $\alpha = \pi$: $\text{Re}(z) = -1 - 1 = -2$, $\text{Im}(z) = 0$, so $z = -2$
Step 4° · $z = a + ib + c + i$
Group all real terms together and all imaginary terms together (assuming $a, b, c$ are real):
$$ z = a + ib + c + i = (a + c) + i(b + 1) $$
$$ \boxed{\text{Re}(z) = a + c, \quad \text{Im}(z) = b + 1} $$
Note: both parts depend on the parameters $a$, $b$, $c$.
Verification: $(a + c) + i(b + 1) = a + c + ib + i = a + ib + c + i$ ✓
Summary Table
| Complex number $z$ | Re$(z)$ | Im$(z)$ |
|---|---|---|
| $-7i$ | $0$ | $-7$ |
| $2 - 4i$ | $2$ | $-4$ |
| $\cos\alpha + i\sin\alpha - 1$ | $\cos\alpha - 1$ | $\sin\alpha$ |
| $a + ib + c + i$ | $a + c$ | $b + 1$ |
Key Points to Remember
- The imaginary part is a real number (the coefficient of $i$), not including $i$ itself.
- To find real and imaginary parts: group all terms without $i$ (real part), then group all coefficients of $i$ (imaginary part).
- For $z = x + iy$: $\text{Re}(z) = x$ and $\text{Im}(z) = y$ (NOT $iy$).
- Both real and imaginary parts can be functions of parameters (like $\alpha$, $a$, $b$, $c$).
Example: for $z = 3 + 5i$, the correct reading is $\text{Re}(z) = 3$, $\text{Im}(z) = 5$ — not $\text{Im}(z) = 5i$.
Result
General principle: write $z$ in standard form $x + iy$; the real part is the sum of terms without $i$, and the imaginary part is the sum of coefficients of $i$.