Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Quaternions
Name: bryan a bertoglio
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995

Could someone explain quaterions to me? When working with fractals, I noticed a puzzling set of rules that they follow:

1: i^2 = j^2 = k^2 = l^2 = -1

2: sorry, I forgot, but I think i * j = -j * i or something weird It had to do with plotting fractals in these four dimensions, fixing one, a and making 3-D fractals. I could not picture any way the rules could work, and my calculus teacher had never heard of them. If you need the other rule, I will post it later (x*y is not equal to y*x for these . . . how?)

The system of quaternions is an extension of the complex number system in the same way that the complex number system is an extension of the real number system. Recall that you treat 2 + 3i algebraically just as you would treat 2 + 3x except that when multiplying you replace any occurrence of i*i by -1. You would treat a quaternion such as 2 + 3i + 5j - 2k algebraically just as you would treat 2 + 3x + 5y - 2z except that you replace products of i, j, and k according to the following rules:

i * i = -1, j * j = -1, k * k = -1, i * j = k, j * k = i, k * i = j,

j * i = -k, k * j = -i, and i * k = -j.

As you observed, this does make multiplication not commutative which may seem weird, but is not any weirder than the i * i = -1 in the complex numbers or the non-commutativity of matrix multiplication.


Also, if you are familiar with cross products and dot products, if you represent a 3-dimensional vector (x, y, z) by the quaternion x i + y j + z k, then the product of two such 3-D vectors, using the quaternion multiplication rules gives a quaternion A i + B j + C k + D, where the 3-D vector (A,B,C) is the cross product of the two original vectors, and D is the negative of their dot product. So quaternion multi- plication actually is slightly related to three-dimensional vector opera- tions.


Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (, or at Argonne's Educational Programs

Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory