In order to calculate the square of any real number, there must be some way to calculate the square root of a negative number. That is, there must be a number such that when it is squared, it is equal to a negative number. For this reason, mathematicians defined what is called the number *i*. **The number *i* is a number such that *i² = -1*.* The number *i* is also called the imaginary identity.**3. If *i² = -1*, then what is the value of *i*?****4. Recall the function *p(x) = x² + 1*. Write the zeros of the function in terms of *i*.**Functions and equations that have solutions requiring *i* have imaginary zeros or imaginary roots.**5. How can you tell from the graph of a quadratic equation whether or not it has real solutions or imaginary solutions?****6. Do you think you can determine the imaginary solutions by examining the graph? Explain your reasoning.**---**Activity 6.2**The set of complex numbers is of the form *a + bi*, where *a* and *b* are real numbers. Any number in the form of a complex number can be expressed as *a + 0i* (a real number) or *0 + bi* (a purely imaginary number). **Diagram Explanation:**The page includes a diagram showing the hierarchy of number systems. It features a flowchart with categories:- **Rational Numbers (Q):** Includes integers, which further include whole numbers, and finally natural numbers, indicating subsets.- **Integers (Z):** Whole and natural numbers are subsets.- **Whole Numbers (W):** Natural numbers are a subset.- **Natural Numbers (N):** The most basic subset.This chart illustrates the relationship and categorization of numbers, emphasizing how complex numbers encompass real and imaginary parts.

given some expression quadratic equation

4. Recall the function p(x) = x? + 1. Write the zeros of the function The number i is a number such the imaginary identity. For this reason, The s - i is e number gh they mbers, al enough s very 3. If i? = -1, then what is the value of i? a + b comp numl The s form of im. num in terms of i. equal 20 Functions and equations that have solutions requiring i have imaginary zeros or imaginary roots. 5. How can you tell from the graph of a quadratic equation whetl or not it has real solutions or imaginary solutions? Rat ning, Inc.

4. Recall the function p(x) = x? + 1. Write the zeros of the function The number i is a number such the imaginary identity. For this reason, The s - i is e number gh they mbers, al enough s very 3. If i? = -1, then what is the value of i? a + b comp numl The s form of im. num in terms of i. equal 20 Functions and equations that have solutions requiring i have imaginary zeros or imaginary roots. 5. How can you tell from the graph of a quadratic equation whetl or not it has real solutions or imaginary solutions? Rat ning, Inc.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

In order to calculate the square of any real number, there must be some way to calculate the square root of a negative number. That is, there must be a number such that when it is squared, it is equal to a negative number. For this reason, mathematicians defined what is called the number *i*. **The number *i* is a number such that *i² = -1*.* The number *i* is also called the imaginary identity.

**3. If *i² = -1*, then what is the value of *i*?**

**4. Recall the function *p(x) = x² + 1*. Write the zeros of the function in terms of *i*.**

Functions and equations that have solutions requiring *i* have imaginary zeros or imaginary roots.

**5. How can you tell from the graph of a quadratic equation whether or not it has real solutions or imaginary solutions?**

**6. Do you think you can determine the imaginary solutions by examining the graph? Explain your reasoning.**

---

**Activity 6.2**

The set of complex numbers is of the form *a + bi*, where *a* and *b* are real numbers. Any number in the form of a complex number can be expressed as *a + 0i* (a real number) or *0 + bi* (a purely imaginary number).

**Diagram Explanation:**

The page includes a diagram showing the hierarchy of number systems. It features a flowchart with categories:

- **Rational Numbers (Q):** Includes integers, which further include whole numbers, and finally natural numbers, indicating subsets.
- **Integers (Z):** Whole and natural numbers are subsets.
- **Whole Numbers (W):** Natural numbers are a subset.
- **Natural Numbers (N):** The most basic subset.

This chart illustrates the relationship and categorization of numbers, emphasizing how complex numbers encompass real and imaginary parts.

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