Question

c. which venn diagram below accurately represents the two sets of numbers? explain.

1. why does multiplying \\(5+7i\\) by \\(5-7i\\) result in a real number? explain.

Explanation:

Analyze the relationship between real and complex numbers

$$ \mathbb{R} \subset \mathbb{C} $$

Every real number \(a\) can be written as a complex number \(a + 0i\). Therefore, the set of Real numbers is a subset of the set of Complex numbers. The third Venn diagram (with "Real" inside "Complex") represents this relationship.

Calculate the product of the complex conjugates

$$ (5 + 7i)(5 - 7i) = 5^2 - (7i)^2 = 25 - 49i^2 $$

Since \(i^2 = -1\):

$$ 25 - 49(-1) = 25 + 49 = 74 $$

The product of a complex number and its conjugate always results in a real number because the imaginary terms cancel out.

Answer:

Question 1

The third Venn diagram (with the "Real" circle entirely inside the "Complex" circle) is correct because every real number is a subset of complex numbers (where the imaginary part is zero).

Question 2

Multiplying complex conjugates \( (a+bi)(a-bi) \) results in \( a^2 + b^2 \). For \( (5+7i)(5-7i) \), the product is \( 5^2 - (7i)^2 = 25 - 49(-1) = 74 \), which is a real number because the imaginary parts cancel out.