We usually count 0,1,2,3,4,5,6,7,8,9, 10, 11, 12…… and we view the number 134 as being made of 1 hundred, 3 tens and 4 units; hence we use expressions like indicating that the tens digit in 134 is 3. We are not constrained to always count in tens (we often don’t on a clock) and this sheet explores some of the easier situations.
Let us begin with the easiest counting system; it would help if you remember me teaching you to count on your fingers. A finger is either up or down; the digit is either 0 or 1. This is base two. So 1101 has four digits; eight, four, (no two) and one. I write this as 11012 = 1310 that is, thirteen. Similarly 1101012= (32+16+4+1)10=5310
Exercise:
1 Convert each of these back to base ten (called denary): 101101, 1111, 111011, 1010101.
2 Convert these to base two (called binary) from denary: 15, 30, 29, 58, 51, 102, 204
3 If I tell you that 1112 is seven, write down the binary form of 1410, 2810, 5610, 11210, 22410.
The last exercise should have persuaded you that doubling is easy in base two (just as multiplying by ten is easy in base ten). So let us look at the process of addition: I have shown the carrying figures….. and one for you to do. The figures sometimes do not line up because TAB doesn’t copy well. I have replaced tabs with spaces - still not satisfactory, but less bad on some browsers
1 1 1 0 1 + 1 0 0 1 1 1 0 1 0 1 1 + 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0
1 0 1 0 1 1 0 0 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 1 1
1 1 1 1 . 1 1 1 1 1 1 1 1 1 . . . . . . .
1 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 . . .
The numbers look large, but are not: the last calculation is 2193010 + 1457910 = 3650910.
Subtraction is a little harder, and you may have difficulty following my carry figures; I’m tempted to skip this, so I’ll settle for just two examples - and one for you to do - as with addition.
1 1 . 1 1 1 1
1 1 1 1 0 1 _ 1 0 0 1 1 1 0 1 0 1 1 _ 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 _
1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 1 1
1 1 . 1 1 1 . . . . . . .
1 0 1 1 1 0 1 0 0 1 0 1 1 0 0 0 1 0 . . .
You may have tried to do these on your calculator; the numbers are too long to manage all but the first. However, if you read the digits in sets of three (from right to left) you can convert rapidly to base eight. The conversions for one to seven go like this: 001, 010, 011, 100, 101, 110, 111, This makes the first of these to be 758 – 278 and the third 526528 – 343638. Your calculator may do these and may even handle the conversions; you may be able to handle the subtraction in base eight (called octal) yourself. I leave the second one for you to convert and use as practice. The point of this work is not to persuade you to upgrade your calculator, it is to establish that you understand the arithmetic processes (e.g. addition) well enough to function in other bases.
Multiplication and division are relatively hard. To finish this page here are more additions and subtractions, in strange bases. If ‘normal’ counting is base 10 and I write it as denary10, then the usual examples would be written binary2, ternary3, quartenary4, octal8, duodecimal12, hexadecimal16. There are names for the intermediate bases, but they are rarely seen.
Exercise: each question has four parts
1 Add 1253 and 3445 in bases 10, 8 and 6. Why can you not add these in base five?
2 Add 2 210 221 and 1 222 012 in ternary and quartenary. Attempt the subtraction too.
3 Convert 34478 to all other even-numbered bases from 10 to 2.
4 Convert 101 111 011 001 1102 to bases, 4 and 8 (fairly easy) and 3 and 9 (fairly hard).
