Recently, I read the “imaginary number, has the form of:

where the real number a is called real part and the real number b is called dual part.

Arithmetic operations
Dual number can perform the arithmetic operations as below:




Finding derivative using Dual Number
The interesting part of dual number is when it is applied to Taylor Series. When substituting a dual number into a differentiable function using the Taylor Series:

This gives a very nice property that we can find the first derivative, f’(a), by consider the dual part of f(a+bε), which can be evaluated using dual number arithmetic.
For example, given a function

we want to find the first derivative of f(x) at x = 2, i.e. f‘(2). We can find it by using dual number arithmetic where f‘(2) will equals to the dual part of  f(2+ε) according to Taylor Series.

Therefore, f‘(2)= 8/9, you can verify this by finding f’(x) and substitute 2 into it, which will give the same answer.

By using dual number, we can find the derivative of a function using dual arithmetic. Hence, we can also find the tangent to an arbitrary point, Dual Numbers: Simple Math, Easy C++ Coding, and Lots of Tricks” by Gino van den Bergen in GDC Europe 2009.