# What is Not a Real Number: Understanding Complex and Imaginary Numbers

Have you ever heard someone say “that’s not a real number?” Well, they might actually be right. As it turns out, there are certain values in math that aren’t considered real numbers. These non-real numbers are a subset of what is known as complex numbers, which is a fascinating field of math that delves into the imaginary realm.

One of the most well-known non-real numbers is a value called “i,” which stands for imaginary. It’s defined as √-1, which means that when you square it, you get negative one. While it might seem odd to use a value that involves the square root of a negative number, imaginary numbers have become ubiquitous in physics, engineering, and various other fields.

While imaginary numbers might seem like something out of a science fiction novel, they’re actually critical to our understanding of the world around us. They allow us to model complex physical systems and solve problems that would be impossible with standard real numbers alone. So, next time someone tells you that a value isn’t a real number, you can respond with a smile, knowing that you’re just exploring the wonderful world of complex numbers.

## Definition of Real Numbers

Real numbers are the set of all rational and irrational numbers that can be plotted on a real number line. But not all numbers can be considered real numbers.

## The Number 1

The number 1 is considered a real number as it can be represented on a real number line. However, there are certain numbers that are not considered real numbers, and one of them is infinity. Infinity is not a real number because it does not have a definitive value; it is merely a concept used to describe an unbounded quantity. For example, if you divide any number by 0, the result is infinity, but infinity itself is not a real number.

• Another number that is not considered a real number is imaginary numbers.
• These are numbers that are expressed in terms of “i,” the imaginary unit, which is equal to the square root of -1.
• These numbers cannot be plotted on a real number line, but they are incredibly important in fields such as engineering, physics, and mathematics.

In addition to imaginary numbers, there are also complex numbers, which are a combination of real and imaginary numbers. Complex numbers can be represented on a two-dimensional plane called the complex plane, with the real number line as the horizontal axis and the imaginary number line as the vertical axis.

Number Type Description
Natural Numbers Positive integers (1, 2, 3, …)
Whole Numbers Natural numbers plus 0 (0, 1, 2, 3, …)
Integers Whole numbers and their negatives (-3, -2, -1, 0, 1, 2, 3, …)
Rational Numbers Numbers that can be expressed as a fraction (0.5, 0.75, 1.25, -2/3, etc.)
Irrational Numbers Numbers that cannot be expressed as a fraction (pi, e, sqrt(2), etc.)
Real Numbers The set of all rational and irrational numbers that can be plotted on a real number line

Understanding what is and is not a real number is important in many areas of study, including science, engineering, and economics. By grasping the concepts behind these number sets, we can better understand the world around us and make accurate calculations and predictions.

## Rational Numbers vs. Irrational Numbers

When it comes to numbers, there are two main categories: rational numbers and irrational numbers. Rational numbers are those that can be expressed as a ratio of two integers, while irrational numbers cannot be expressed in this way. Here, we will dive into what is not a real number, focusing on the number 2.

## The Number 2

At first glance, it may seem like 2 is a rational number. After all, we can express it as the ratio of 2 and 1. However, 2 is not just a rational number. In fact, it is not a real number at all. This may sound strange, but it is because 2 is an imaginary number.

Imaginary numbers are numbers that can be expressed in the form of a real number multiplied by the square root of -1. In the case of 2, we can write it as 2i, where i is the imaginary unit or the square root of -1. This may seem like a bizarre concept, but imaginary numbers play a crucial role in mathematics, particularly in complex numbers and the study of waves and vibrations.

## Wrapping Up

In summary, 2 is not a real number, but rather an imaginary number expressed as 2i. While this may seem confusing at first, it highlights the importance of understanding different types of numbers in mathematics. It also showcases the incredible complexity and beauty of numbers as a whole.

## Imaginary Numbers

Imaginary numbers are a subset of complex numbers that cannot be represented on the real number line. Unlike real numbers, which can be positive, negative, or zero, imaginary numbers are always expressed in terms of the imaginary unit “i,” where i = √(-1).

One common misconception about imaginary numbers is that they are not “real.” However, this is not entirely true. Although imaginary numbers cannot be represented on the real number line, they still play an important role in mathematics and are used to solve a variety of complex problems.

## The Number 3

The number 3 is a real number and is not considered an imaginary number. This is because it can be represented on the real number line and is not expressed in terms of the imaginary unit “i.”

To further understand the concept of imaginary numbers, it may be helpful to review the basics of complex numbers. A complex number is expressed in the form a + bi, where a and b are both real numbers and i is the imaginary unit. For example, 2 + 3i is a complex number.

• Real Numbers: positive, negative, or zero numbers that can be represented on the real number line.
• Imaginary Numbers: expressed in terms of the imaginary unit “i” and cannot be represented on the real number line.
• Complex Numbers: a combination of a real number and an imaginary number expressed in the form a + bi.

When performing mathematical operations on complex numbers, it is important to keep in mind the rules for working with imaginary numbers. For example, multiplying two imaginary numbers together will always result in a real number.

Operation Result
i x i -1
i x 2i -2
(2 + 3i) x (4 – i) 11 + 10i

It can also be helpful to plot complex numbers on the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This can make it easier to visualize complex mathematical concepts and solve complex problems.

## Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. The imaginary part is a multiple of the imaginary unit “i” (which is defined as the square root of -1). Complex numbers are often written in the form a+bi, where a is the real part and b is the imaginary part.

• The set of complex numbers does not include any numbers that do not have an imaginary part. For example, the number 4 is not a complex number because it does not have an imaginary part. Instead, it is considered a real number.
• There are several arithmetic operations that can be performed on complex numbers, including addition, subtraction, multiplication, and division. These operations follow certain rules, such as the distributive property and the fact that the product of two imaginary units is -1.
• One common application of complex numbers is in the study of electrical engineering and physics, where they are used to model circuits and waves. They are also used in many other areas of science, such as quantum mechanics.

One interesting property of complex numbers is that they can be plotted on a complex plane, where the real part is represented on the horizontal axis and the imaginary part is represented on the vertical axis. This allows for a visual representation of complex numbers and their relationships to one another.

Operation Expression Result
Addition (a+bi) + (c+di) (a+c) + (b+d)i
Subtraction (a+bi) – (c+di) (a-c) + (b-d)i
Multiplication (a+bi)(c+di) (ac-bd) + (ad+bc)i
Division (a+bi) / (c+di) ((ac+bd)/(c^2+d^2)) + ((bc-ad)/(c^2+d^2))i

Overall, complex numbers are a fascinating and useful topic in mathematics with a wide range of applications in various fields.

## Non-numeric Values

When we talk about real numbers, we are referring to the set of numbers that can be represented on a number line. However, not all values that we commonly encounter in our lives are considered real numbers. These values are known as non-numeric values, and they include things like infinity, NaN, and imaginary numbers.

## NaN (Not a Number)

• NaN is a value that represents an undefined or unrepresentable value.
• It is often the result of an invalid mathematical operation, such as dividing by zero or taking the square root of a negative number.
• NaN is not equal to anything, not even itself.

## Infinity

Infinity is a value that represents a number that is larger than any finite number. It can be represented by the symbol ∞. There are two types of infinity: positive infinity and negative infinity.

• Positive infinity is represented by the symbol ∞ and is used to represent an infinitely large positive value.
• Negative infinity is represented by the symbol -∞ and is used to represent an infinitely large negative value.

## Imaginary Numbers

Imaginary numbers are a type of complex number that cannot be represented on a number line. They are expressed as a multiple of the square root of -1, which is represented by the symbol √-1.

Real Part Imaginary Part Example
5 2i 5 + 2i
0 -3i -3i

Imaginary numbers are used in many areas of mathematics and science, including electrical engineering and quantum mechanics.

In conclusion, understanding non-numeric values is important in many areas of study, including mathematics, science, and engineering. NaN, infinity, and imaginary numbers are all examples of values that cannot be represented on a standard number line, but they play a crucial role in many important mathematical and scientific concepts.

## Infinity as a Concept

Infinity is a concept that has fascinated mathematicians and scientists for centuries. But what exactly is infinity, and is it really a number? Contrary to popular belief, infinity is not a real number. It cannot be represented by any finite value, and as such, it is often used as a placeholder or symbol to represent something that is unbounded or never-ending. However, there is much more to this concept than meets the eye. Let’s take a closer look at what infinity really is and how it relates to the concept of numbers.

## The Number 6: Is It Finite or Infinite?

When we think of numbers, we typically think of them as either finite (having a fixed and definite value) or infinite (not bounded by any limit or ending). But what about the number 6? Is it finite or infinite? The answer might surprise you.

• First and foremost, the number 6 is a finite number. It has a fixed and definite value, which is 6.
• However, the number 6 can also be thought of as an infinite number, depending on how we define it. For example, we can write the number 6 as a repeating decimal: 6.000000… (with the zeroes repeating infinitely). In this sense, we can say that 6 is an infinite number.
• Furthermore, we can think of 6 as a member of an infinite set of numbers, such as the set of natural numbers (1, 2, 3, 4, 5, 6, …), which goes on infinitely.

So while the number 6 is technically a finite number, it is also closely related to the concept of infinity and can be thought of as both finite and infinite depending on the context in which it is being used.

## Prime Numbers as a Subset of Real Numbers

Prime numbers are the building blocks of the natural numbers. They are positive integers that are only divisible by 1 and themselves. However, not all natural numbers are prime. Some natural numbers, like 4 and 6, can be factored into smaller integers. These are called composite numbers.

But prime numbers are not the only types of numbers that exist. In fact, there are numbers that are not considered real numbers at all. These numbers are often denoted as complex numbers or imaginary numbers.

## The Number 7

• 7 is a prime number. It is not divisible by any integer other than 1 and itself.
• 7 is also a natural number. It can be expressed as a counting number or a whole number.
• 7 is not an irrational number. It can be expressed as a fraction, 7/1.

However, 7 is not a complex number. Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is equal to the square root of -1.

For example, the number 3 + 4i is a complex number. It has a real part of 3 and an imaginary part of 4i. But there is no imaginary part to the number 7, so it cannot be expressed as a complex number.

Type of Number Examples
Real Numbers 1, 2/3, -6, Pi
Imaginary Numbers 5i, -2i/3, i
Complex Numbers 3 + 4i, -2 – 5i, 1/2 + i

While 7 may seem like a simple number, it is a prime number that cannot be expressed as a complex number. Understanding what types of numbers exist and their properties is important in mathematics and in real-life applications.

## FAQs: What is Not a Real Number?

Q: What is a real number?

A: A real number is any number that can be found on the number line, including positive, negative, and decimal numbers.

Q: What is not a real number?

A: Anything that cannot be plotted on the number line is not a real number.

Q: Are complex numbers real?

A: No, complex numbers are not real. They consist of a real part and an imaginary part and cannot be plotted on the number line.

Q: Are fractions real numbers?

A: Yes, fractions are real numbers. They can be plotted on the number line as decimals or between whole numbers.

Q: Is pi a real number?

A: Yes, pi is a real number. It is a non-repeating, non-terminating decimal that can be plotted on the number line.

Q: Are negative numbers real?

A: Yes, negative numbers are real numbers. They can be plotted to the left of zero on the number line.

Q: Is infinity a real number?

A: No, infinity is not a real number. It is not a specific value, but rather a concept representing an unbounded quantity.

## Closing Thoughts

Thank you for reading and learning about what is not a real number. Remember that any number that cannot be plotted on the number line is not a real number, including complex numbers and infinity. Fractions, decimals, and negative numbers are all examples of real numbers. Visit us again for more interesting and informative articles.