Base Fundamentals

Just when you thought you can do arithmetic….

We tend to count in tens – largely because we have ten fingers. Some languages work in fives, so that six is ‘five and one’. Consider the number 123; we generally mean one hundred, two tens and three digits. That’s in base ten, called denary.

Try the same number, 123, in base 5; it’s still three ones, but it has two fives and one twenty five. Why?

To make these numbers easier (or less difficult) to follow, I shall write a subscript showing the base, e.g. 1235  .

Adding should not be too difficult: 1215  + 1235  =  2445 is easy because there are no carry figures, but you’ll have to think about 123+ 1235  which is two twenty fives. four fives and six units. The six units become a five and one, so we now have five fives, or an extra twenty five, that is, three of them, so 123+ 1235 = 3015capisce?

You try:       3215   + 3215     241235  + 401455      14567+ 60567      

                    3829 + 5739     10111+ 11011012        38211 + 57311

So in any old base, say base n, the right-most column is digits (ones) the next column is n, the next going left is n2 and so on. We put the biggest term on the LEFT, so it reads ……. n4 n3 n2n1 n0 and you just might have an “animal’ point, so it continues      n4 n3 n2 n1 n0. n-1 n-2 n2 n-3n-4…..

This is why I write my dates with the most significant part on the left. Today is 20051213, the 13th December 2005. Right now is 20051213 133704: this is yyyymmdd hhmmss format, or Standard European Date/Time format – and I don’t need the : separators that I would usually insert for the time 13:37:04.

Some of the bases have names; bases 2,3,4,8,10,12,16 are called binary, ternary, quartenary, octal, denary, duodecimal, hexadecimal. Your calculator probably can be persuaded to work in bases 2, 8, 10, and 16. For twelve and sixteen you need extra characters; 1,2,3,4,5,6,7,8,9,t,e (ten eleven) for base twelve and …7,8,9,A,B,C,D,E,F in base sixteen. So FF is fifteen sixteens plus fifteen ones, FF16 = FF H = 25510.

Try these additions and subtractions:

1)  Add 234 and 456 in bases 7, 9 and 12.

2)  Add 12022 to 21022 in bases three, and four and five - that’s 3 problems.

3)  Take 23041 from 44032 in base seven. Do it again but read them as being in base five. Translate the last problem to denary.

4)   Add 1101, 1101, 1101 in binary. Compare this with 11012 x 112 and do 11012 x 1102

  1. 5)   Take 100112 from 11101102.  Show that this translates to 13124 -  1034 and  1668 - 238.

I like “animal point”, but some will say I should call it an anymal point.



© David Scoins 2017