Final answer:
To add the complex numbers (7+2i) and (1−4i), combine the real parts (7+1) to get 8, and combine the imaginary parts (+2i−4i) to get −2i. The sum in standard form is 8−2i.
Step-by-step explanation:
To add the complex numbers (7+2i) and (1−4i), we combine the real parts and the imaginary parts separately. When we add two numbers with the same sign, we simply combine their magnitudes and keep their common sign. When we combine numbers with opposite signs, we subtract the smaller magnitude from the larger one and keep the sign of the number with the larger magnitude.
Combining the real numbers, we have:
7 (from the first complex number) + 1 (from the second complex number) = 7 + 1 = 8
Combining the imaginary numbers, we get:
+2i (from the first complex number) − 4i (from the second complex number). Since these have opposite signs, we subtract and keep the sign of the larger absolute value:
2i − 4i = −2i (note that − 4i is larger in absolute value, so we keep the negative sign)
Adding the result of the real parts to the result of the imaginary parts gives us:
8 + (−2i) = 8 − 2i
So, the complex number in standard form after addition is 8 − 2i.