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Chapter 1: Problem 6

True or False All real numbers are complex numbers.

Short Answer

Expert verified
True, all real numbers are complex numbers because they can be written as \(a + 0i\).

Step by step solution

01

Define Real Numbers

Real numbers include all the numbers on the number line. They can be positive, negative, or zero. Examples include \(1, -3, 2.5, \sqrt{2}\).
02

Define Complex Numbers

Complex numbers are numbers in the form \(a + bi\), where \a\ and \b\ are real numbers and \i\ is the imaginary unit, defined as \i^2 = -1\. For example, \(3 + 4i\).
03

Identify Relationship Between Real and Complex Numbers

Every real number is a complex number with \(b = 0\), meaning it can be written as \(a + 0i\). For example, \5\ is a real number and can be written as \(5 + 0i\).
04

Conclusion

Based on the definitions and the relationship, we can conclude that all real numbers are indeed complex numbers. Therefore, the statement is True.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Numbers
Real numbers include all the numbers you can find on the number line. They are very familiar to us because they include everyday numbers. Real numbers can be:
  • Positive numbers like 1, 2, or 100
  • Negative numbers like -1, -2.5, or -100
  • Whole numbers like 0, 4, or 10
  • Fractions like 1/2, -3/4, or 0.75
  • Irrational numbers like 蟺 (Pi) and \(\sqrt{2}\)
Real numbers also obey the basic rules of arithmetic, such as addition, subtraction, multiplication, and division (except division by zero).
Imaginary Unit
The imaginary unit, commonly represented as \(i\), is a special number that is defined by its unique property:
\(i^2 = -1\)
This means that if you square the imaginary unit, you get -1. This property distinguishes \(i\) from all real numbers, because squaring any real number always gives a positive result (unless the number is zero). Imaginary units allow us to extend the concept of numbers and solve equations that have no solutions among the real numbers. For instance, the equation \(x^2 + 1 = 0\) is impossible to solve with real numbers, but using imaginary numbers, we can write the solutions as \(x = i\) and \(x = -i\).
Relationship Between Real and Complex Numbers
Complex numbers extend the idea of real numbers by including the imaginary unit. A complex number is written in the form \(a + bi\), where:
  • \(a\) is the real part
  • \(bi\) is the imaginary part
  • \(i\) is the imaginary unit (\(i^2 = -1\))
Through this, we see that complex numbers include all real numbers. This is because any real number can be written as a complex number with an imaginary part of zero. For example, the real number 5 can also be interpreted as the complex number \(5 + 0i\). This relationship shows that all real numbers are also complex numbers, but not all complex numbers are real numbers. Therefore, the statement 'All real numbers are complex numbers' is \text{True}\.

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