The inequality \(2 + 2j \leq 10\) involves a complex number, \(2 + 2j\), where \(j\) is the imaginary unit.

In the context of inequalities involving complex numbers, complex numbers are **not** ordered in the same way real numbers are. Inequalities like \(z_1 \leq z_2\) (where \(z_1\) and \(z_2\) are complex numbers) are not generally defined because complex numbers do not have a natural ordering.

However, you can interpret this inequality as a comparison of **magnitudes** (or absolute values). The magnitude (or modulus) of a complex number \(z = a + bj\) is given by:

\[
|z| = \sqrt{a^2 + b^2}
\]

So, for \(2 + 2j\), the magnitude is:

\[
|2 + 2j| = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.83
\]

Thus, interpreting the inequality \(2 + 2j \leq 10\) as comparing the magnitude, we check if \(2\sqrt{2} \leq 10\), which is true because \(2.83 \leq 10\).

### Summary:
The inequality \(2 + 2j \leq 10\) holds **if interpreted as a magnitude comparison**.

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. What is the modulus of the complex number \(3 + 4j\)?
2. Can you compare two complex numbers using inequalities? Why or why not?
3. How do you represent a complex number in polar form?
4. What is the argument of a complex number, and how do you calculate it?
5. What does it mean for two complex numbers to be equal?

**Tip**: Complex numbers are best compared by their modulus (magnitude) when thinking about sizes, rather than using inequality symbols.

Explore the inequality involving complex numbers: 2 + 2j ≤ 10. Learn how to interpret this inequality by comparing the magnitude of the complex number 2 + 2j, and check if its modulus is less than or equal to 10. This problem covers concepts of complex numbers, magnitude, and inequalities, and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation