s the work shown below correct? Explain your answer. (11 + 2i ) – (3 – 10i ) = 11 + 2i – 3 – 10i = (11 – 3) + (2i – 10i) = 8 – 8i
The Correct Answer and Explanation is:
Given Expression:
(11+2i)−(3−10i)(11 + 2i) – (3 – 10i)(11+2i)−(3−10i)
Step-by-step Simplification:
- Distribute the minus sign:
(11+2i)−3+10i(11 + 2i) – 3 + 10i(11+2i)−3+10i
Note that subtracting a negative number becomes addition: 11+2i−3+10i11 + 2i – 3 + 10i11+2i−3+10i
- Combine like terms:
- Real parts: 11−3=811 – 3 = 811−3=8
- Imaginary parts: 2i+10i=12i2i + 10i = 12i2i+10i=12i
Correct Answer:
8+12i8 + 12i8+12i
What Went Wrong in the Given Work:
The work shown was: (11+2i)–(3–10i)=11+2i–3–10i=(11–3)+(2i–10i)=8–8i(11 + 2i) – (3 – 10i ) = 11 + 2i – 3 – 10i = (11 – 3) + (2i – 10i) = 8 – 8i(11+2i)–(3–10i)=11+2i–3–10i=(11–3)+(2i–10i)=8–8i
This simplification incorrectly handled the subtraction of the complex number (3−10i)(3 – 10i)(3−10i). The minus sign should have been distributed to both terms inside the parentheses, turning −(3−10i)- (3 – 10i)−(3−10i) into −3+10i-3 + 10i−3+10i. However, in the original work, the sign of −10i-10i−10i was mistakenly kept negative instead of being flipped to +10i+10i+10i.
Because of this error, the imaginary part was incorrectly calculated as: 2i−10i=−8i(Incorrect)2i – 10i = -8i \quad \text{(Incorrect)}2i−10i=−8i(Incorrect)
Instead of: 2i+10i=12i(Correct)2i + 10i = 12i \quad \text{(Correct)}2i+10i=12i(Correct)
Explanation
When subtracting complex numbers, it’s essential to treat the subtraction like you would with algebraic expressions involving parentheses. Each component of the complex number (the real part and the imaginary part) must be subtracted individually. In this case, the expression involves subtracting the complex number (3−10i)(3 – 10i)(3−10i) from (11+2i)(11 + 2i)(11+2i).
The mistake in the original work occurred because the person did not correctly apply the minus sign to both terms inside the second complex number. Subtracting a complex number is the same as distributing the negative sign to both its real and imaginary parts. So, (11+2i)−(3−10i)(11 + 2i) – (3 – 10i)(11+2i)−(3−10i) becomes 11−3+2i+10i11 – 3 + 2i + 10i11−3+2i+10i.
Once this is done, the real parts and the imaginary parts can be combined separately:
- The real parts: 11−3=811 – 3 = 811−3=8
- The imaginary parts: 2i+10i=12i2i + 10i = 12i2i+10i=12i
Thus, the final and correct result is: 8+12i\boxed{8 + 12i}8+12i
This kind of error is common when students rush or don’t carefully apply the rules of signs. Always remember: parentheses and sign distribution matter, especially when working with expressions that include negative values or subtraction. Proper use of these rules ensures accurate simplification of complex numbers.
