# Approximation of Numbers

Numbers produced by a calculator are often inexact because the calculator can work only with a finite number of decimal places. For instance, a $10$-digit calculator gives $2 \div 3 = 6.666666667 \times {10^{ - 1}}$ and $\sqrt 2 = 1.414213562$, both of which are approximations of the true values.

Don’t be too quick to pick up your calculator –answers such as $2/3$, $\sqrt 2$, $(\sqrt 2 + \sqrt 3 )/7$ and $\pi /4$ are often preferred to more lengthy decimal expressions that are only approximations.

Most numbers obtained from measurements of real-world quantities are subject to error and also have to be regarded as approximations. If the result of a measurement (or any calculation involving approximations) is expressed in scientific notation, $p \times {10^n}$, it is usually understood that $p$ should contain only significant digits, that is, digits that, expect possibly for the last, are known to be correct or reliable. The last digit may be off by one unit because the number was rounded off.

For instance, if we read in a physics that one electron volt $= 1.60 \times {10^{ - 19}}$ joule we understand that the digit $1$, $6$ and $0$ are significant and we say that, to an accuracy of three significant digits, one electron volt is $1.60 \times {10^{ - 19}}$ joule.

To emphasize that a numerical value is only an approximation, we often use a wave-shaped equal sign $\approx$. For instance, $\sqrt 2 \approx 1.414$

However, we sometimes an ordinary equal sign when dealing with inexact quantities simply because it becomes tiresome to repeatedly indicate that approximations are involved.