Sources of Error

# Sources of Error

It is always important to acknowledge possibly sources of error - especially when it comes to applied mathematics dealing with biology, physics, chemistry, engineering, economics, etc… We will now outline some of the sources of error.

**Calculation Errors**. This is the most obvious type of error that arises as a result of an incorrect calculation. For example, $1 + 1 = 3$. Other types of calculation errors can result due to assumptions.

**Rounding/Truncation Errors**. We have already discussed this type of error on the Truncation of Floating-Point Numbers and Rounding of Floating-Point Numbers pages. These sort of errors arise frequently when an exact value cannot be used directly, for example, using $3.14$ to represent $\pi$. It is a lot easier to use $3.14$ represent $\pi$, however, for precision, $3.14$ may produce a large amount of error when applied to many of the elementary geometry formulas for area and volume, such as $A = \pi r^2$ (the area formula for a circle) or $V = \pi r^2 h$ (the volume of a right circular cylinder).

**Modelling Errors**. Many phenomena in the world can be represented with mathematical models. Of course, the world is so complex that it would be virtually impossible to create a mathematical model that represented some situation with complete accuracy . As a result, many mathematical models have errors associated to missing variables or to perturbations in the normalcy of a model.

**Approximation Errors**. There are many instances in mathematics where we approximate some value with another value. One common example is $\pi \approx \frac{22}{7}$. Another common example is using power or Taylor polynomial series to represent functions. For example, the geometric series below represents the function $f(x) = \frac{1}{1 - x}$ for $\mid x \mid < 1$:

\begin{align} \quad \sum_{n=0}^{\infty} x_n = 1 + x + x^2 + ... = \frac{1}{1 - x} \end{align}

- Thus we have that $\frac{1}{1 - x}$ is can be approximated by a finite number of consecutive terms (starting at the first term) of the series given above, that is for $\mid x \mid < 1$ we have that:

\begin{align} \frac{1}{1 - x} \approx 1 + x + x^2 + ... + x^n \end{align}