# Quaternions: Extension of Real Numbers?

• Topolfractal
In summary, Quaternions are generalizations of 3-vectors, in the same way as complex numbers are generalizations of 2-vectors. They have four components and are an extension of the real numbers, just as the complex numbers were. Despite initial confusion, it is now clear that quaternions are not just a 3-dimensional version of the real numbers, but a unique and important mathematical concept in their own right.
Topolfractal
Quaternions are generalizations of 3- vectors, in the same as complex numbers are generalizations of 2- vectors. Should quaternions be considered an extension of the real numbers as the complex numbers were?

Topolfractal said:
Quaternions are generalizations of 3- vectors, in the same as complex numbers are generalizations of 2- vectors. Should quaternions be considered an extension of the real numbers as the complex numbers were?
Wouldn't they be considered generalizations of 4-vectors? A quaternion has four components.

Oh I thought quaternions were just adding 1 more part to a complex number. I must be way wrong.

Topolfractal said:
Oh I thought quaternions were just adding 1 more part to a complex number. I must be way wrong.
The "quatern" part comes from Latin, meaning "four times."

A quaternion has the form q = ai + bj + ck + d, where i, j, and k are such that i2 = j2 = k2 = -1.

Topolfractal said:
Oh I thought quaternions were just adding 1 more part to a complex number. I must be way wrong.

You are not alone. My recollection is that Hamilton was trying to add one more dimension to complex number to make a three part number. He wasn't able to find a way to make add just one dimension and have it be an extension of the complex numbers. Quaternions, however, are an extension to the complex numbers.

DrewD said:
You are not alone. My recollection is that Hamilton was trying to add one more dimension to complex number to make a three part number. He wasn't able to find a way to make add just one dimension and have it be an extension of the complex numbers. Quaternions, however, are an extension to the complex numbers.
Thank you, and you for the longest time having not researched quaternions in depth, I always thought they were a 3-d version of the real numbers with three components. After reading the Wikipedia article on quaternion I understand Hamilton's motivation behind the four parts now.

Topolfractal said:
Quaternions are generalizations of 3- vectors, in the same as complex numbers are generalizations of 2- vectors. Should quaternions be considered an extension of the real numbers as the complex numbers were?
Yes, undoubtedly!

## 1. What are quaternions and how are they different from real numbers?

Quaternions are a mathematical concept that extends the idea of real numbers. They are composed of four components (a, b, c, d) and are written in the form a + bi + cj + dk, where i, j, and k are imaginary units. Unlike real numbers, quaternions have a non-commutative multiplication rule, meaning the order in which they are multiplied matters.

## 2. What is the history of quaternions?

Quaternions were first discovered by Irish mathematician Sir William Rowan Hamilton in 1843. He was looking for a way to extend complex numbers and came up with the idea of adding a third imaginary unit, j, to create quaternions. They were initially met with skepticism but gained popularity in the late 19th and early 20th centuries, particularly in the fields of physics and engineering.

## 3. How are quaternions used in modern science?

Quaternions have a wide range of applications in modern science, particularly in fields such as physics, computer graphics, and robotics. They are used to represent rotations in three-dimensional space and are commonly used in 3D animations, video games, and simulations. They are also used in signal processing, control systems, and quantum mechanics.

## 4. Can quaternions be used in place of real numbers?

No, quaternions cannot be used interchangeably with real numbers. While they can be used to represent real numbers (by setting b, c, and d to 0), they have different properties and operations. For example, quaternions do not have a total ordering like real numbers do, meaning they cannot be arranged from least to greatest.

## 5. Are there any challenges or limitations to using quaternions?

One challenge of using quaternions is that they are not as intuitive as real numbers, and can be difficult to visualize. Additionally, their non-commutative multiplication rule can make calculations more complex. They also have limitations in representing certain types of rotations, such as rotations that involve a change in scale. However, in many cases, the benefits of using quaternions outweigh these challenges.

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