# Calculus without Calculus

###### With exams fast approaching, Newman University’s Senior Lecturer and Course Leader in Mathematics, Andrew Toon discuses a simple idea to help our Mathematics students determine the derivative of any function using the algebra of what are called dual numbers instead of using the many rules of differentiation.

“As some of you may be worried about your calculus skills with exams quickly approaching, I thought I would share with you a simple idea to determine the derivative of any function using the algebra of what are called dual numbers instead of using the many rules of differentiation.

“Some of you may be familiar with complex numbers which can be manipulated using the usual rules of algebra with the understanding that , but don’t worry if you are not!

“In a similar way, we can define what are called dual numbers which take the form , where are real numbers but now has the strange property that . Just like the in complex numbers which is understood in terms of its square , we understand and apply dual numbers by requiring whenever it appears.

“For example

“Also, using Taylor series, we have for example

“This simple madness allows us to express any function that has a Taylor series expansion (whose derivatives exist to all orders) in the form:

“From this we can determine which is simply all terms that once we have fully simplified and used the understanding that .

“Let’s consider a couple of examples:

However, using dual numbers (and remembering ) we have

“Comparing with as expected.

using the product rule.

“However, using dual numbers (and remembering ) we have using the above results

.

“Comparing with as expected.

“So, using dual numbers of the form with the understanding that we can reduce differential calculus to the algebra of such numbers! This gives us a powerful, easy and effective way to evaluate derivatives no matter how complex without needing to know all those many rules of differentiation by using algebra only!

“As a final note, you might be wondering how can there be non-zero quantities such that for complex numbers and (remember ) for dual numbers? What you are in fact seeking is a representation of these quantities that makes sense and this is precisely what a mathematics degree will show you, so, do come and join us in this fascinating world!

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