**Introduction**

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:

Multiplication:

Division:

**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.

**Conclusion**

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.*