Calculating Machines --- Abacus --- Napier's bones --- Slide Rule --- Logarithms --- Calculator

Nowadays, nearly anyone who wants a machine to do a calculation uses a calculator. These were originally called electronic calculators, to distinguish them from mechanical calculators, or pocket calculators, to distinguish them from computers!

Do you makes mistakes while using a calculator? Try the following:

- Check each number after you've entered it.
- If you get confused in the middle of a calculation, use a piece of paper to hide the part of the calculation that you haven't go to yet. If you get confused in the middle of a long number, use paper, or a finger, or a pen, to hide, or point to, the digits of the number.
- Check each result by estimating it. Take the first digit of each number and add enough zeroes to make it roughly the right size. Then it is easy enough to estimate the result. (456 x 64) is roughly (400 x 60) or 24000. (3671 + 9642) is roughly (3000 + 9000) or 12000. For a more precise answer, trying rounding up and down alternately. (456 x 64) is roughly (500 x 60) or 30000 (the correct answer is 29184). (3671 + 9642) is roughly (4000 + 9000) or 13000 (the correct answer is 13313). If you get something like -0.0345, then you know you've got it wrong!
- Even if you do not feel like estimating, you can judge if the result looks reasonable. If you add, multiply or divide positive numbers, you don't expect a negative result. If you multiply large numbers, you get an enormous number. If you add two positive numbers, you get a number bigger than the biggest of them, but less than twice as big. If you look at the result and think "That can't be right!", then you have probably made a mistake!
- Try doing the calculation more than once. If you do the same calculation twice the same way, you might make the same mistake, so try doing it a different way the second time. If you are adding up a string of numbers, first add them from top to bottom, write down the result, then add the numbers from bottom to top, and see if you get the same result! Some calculations are harder to reverse. (3 - 2) is not the same as (2 - 3). It
*is*the same as (-2 + 3). - When doing more than one calculation, use the 'C' key before starting the new calculation, to make sure that no number gets left over from the previous one.

Here is some practice in checking if calculations are reasonable:

You would think that all calculators would work in the same way for simple calculations. In fact, they have their own peculiarities. If you calculate **1 + 2 x 3** on the grey calculator above, it will produce the answer **9**. It adds **2** to **1**, then multiplies the whole thing by **3**. But this isn't the way that mathematicians do calculations. They work out all multiplications and divisions first, then do the additions and subtractions after. So **1 + 2 x 3** gives the same answer as **2 x 3 + 1**, which is **7**. You will find that the calculator below does this. It has other helpful features. You can add or subtract a percentage (e.g. 150 + 17.5%). You can reverse the sign on the result or take the reciprocal ('one over ...'), which is often useful when doing calculations. You can display the result as a whole number, or to 2 decimal places (both rounded), and indeed switch between these if you want. Finally, the whole entered calculation is displayed at the bottom, so you can check if you entered something wrong. You can even change this calculation, and click on 'Redo calculation' to re-calculate it. In the bottom display, the reciprocal is shown as 'n' and the divide by '/', as the other symbols are not in the normal character set.

Most computers have a calculator as part of their operating system. To find the PC calculator, click on Start, then Programs, then Accessories, then Calculator. Click on View to choose Standard or Scientific. Try **1 + 2 x 3** on each!

Calculators have a special notation if the numbersget too large or too small. They display significant figures and powers of ten. The calculators on this page don't, as they are designed for teaching people how to use a calculator, or for simple use. Use your computer's calculator or your own for more complicated arithmetic.

© Jo Edkins 2007 - Return to Numbers index