Deep in unexplored territory, Emek and Tara reached a damaged bridge whose repair controls were marked with real and complex values. Three calculations had to be matched with compatible control panels before the bridge could move. The first displayed (4 + 3i) + (2 - 5i). Tara combined real and imaginary parts to obtain 6 - 2i. The second showed (3 + 2i)(3 - 2i). Emek recognized conjugates and simplified the product to 9 - (2i)^2 = 13, a real number. The final calculation was i(5 - 4i), which simplified to 4 + 5i because i^2 = -1. The control instructions required the real result first, followed by the complex result with a negative imaginary part, then the one with a positive imaginary part. Emek and Tara entered 13, 6 - 2i, and 4 + 5i in that order. The bridge mechanisms restarted, restoring the crossing and allowing the expedition to continue.

Challenges

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Challenge 1 of 5

Challenges

Start
Challenge 1 of 4
Hint
Combine real parts with real parts and imaginary coefficients with imaginary coefficients, keeping the sign on -5i.
A student simplifies (4 + 3i) + (2 - 5i) as 6 + 8i. Which explanation best identifies the mistake?
The student should multiply the real parts instead of adding them.The student added 3 and 5 instead of combining the imaginary coefficients as 3 + (-5).The student should replace i with -1 before adding.The student should combine 4 with -5i and 2 with 3i.
Correct! The imaginary coefficients are 3 and -5, so their sum is -2.
Not quite. Pay attention to the negative sign in -5i when combining the imaginary terms.
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Hint
The factors are complex conjugates. Use i squared = -1.
The second bridge calculation is (3 + 2i)(3 - 2i). What integer value results after simplifying completely?
Correct! The conjugate product simplifies to the real number 13.
Try again. Multiply the conjugates and replace i squared with -1.
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Hint
Classify each simplified result by whether it is real only, has a negative imaginary part, or has a positive imaginary part.
The bridge controls require the real result first, then the complex result with a negative imaginary part, and finally the complex result with a positive imaginary part. Which sequence correctly satisfies the instructions?
6 - 2i, 13, 4 + 5i4 + 5i, 6 - 2i, 1313, 4 + 5i, 6 - 2i13, 6 - 2i, 4 + 5i
Correct! 13 is real, 6 - 2i has a negative imaginary part, and 4 + 5i has a positive imaginary part.
Not quite. Match each result to the required type before placing them in order.
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Karmaşık Geçitten Karşıya
TR
ARC_G12_NUM_02_STORY_03
Karmaşık Geçitten Karşıya
Across the Complex Crossing
EN
ARC_G12_NUM_02_STORY_03
Across the Complex Crossing