## Round numbers

It seems that we have a fascination for numbers that end in zeros. One thousand (1,000) and one million (1,000,000) and so on and to a lesser extent numbers like one hundred (100) and ten thousand (10,000). Fractions of round numbers also appeal to people. Reaching the age of 50 (half of 100) or 75 (3/4 of 100) is considered an interesting milestone while reaching 74 or 51 for some reason is not so interesting.

However some fractions are not even noticed. At the age of 33 years and 4 months you will be 1/3 of 100 years old and at 66 years and 8 months you will be 2/3 of 100 years old. When I passed the second milestone, I mentioned it to people and they didn’t seem to care. No prezzies were forthcoming.

There is one special number that could be considered a round number, and the wellspring of all round numbers and that is the number zero. The first number which is usually considered a round number is ten (10), where the zero indicates the absence of any digit (except zero itself, which is normally considered a numeric digit), and the 1 is positional and represents the number ten or ten units. Stick another zero on the end has the effect of multiplying all the digits in the number by ten, so 10 becomes 100 which represents one hundred.

The number 100 could be considered to be, reading from the right, zero ‘units’, zero ‘tens’ and one ‘hundred’. 110 is zero ‘units’, one ‘ten’ and one ‘hundred’ giving the number one hundred and ten. When I was learning arithmetic as a small boy, while I grasped the principles quickly enough, I continued to ponder this mapping process from numbers, like seven hundred and thirty one to the mathematical representation of 731. Maybe I was a strange child. I still ponder it, even today. I look on “seven hundred and thirty one” as a name of the number, ‘731’ as a representation of the number, and the number itself as some ineffable thing. Maybe I grew up to be a strange adult!

Notice that above I said that “seven hundred and thirty one” is a name of the number and ‘731’ is a representation of the number. This is because there can be other representations of the same number, mainly in different bases. For all the numbers above I’ve used the number ten (10) as the base, but I could have used another base, such as sixteen (16) which is frequently used in computers. The number “seven hundred and thirty one” in base 10 representation is represented as ‘2D8’ in base 16 or hexadecimal representation. The ‘D’ is in there to represent the number thirteen (decimal 13).

Any positive integer can be used as a base. The bigger the base the more ‘digits’ are required to represent numbers, making them hard to read and hard to calculate with, so a base of ten (10) is a reasonable choice for general use. Negative integers can also be used as bases, but then things get very difficult! I’ve occasionally wondered if rational numbers or real numbers could be used as bases, but I can’t see how that would work.

Computers are interesting, since they, at the lowest level, appear to use a base of two (2), which is the smallest possible positive integer base. The numbers are conceptually simple strings of ones (1) and zeros (0) called ‘bits’. It’s not as simple as that however as in the computer’s central processor the data and programs are shunted around like little trains of bits, switching from track to track and in many cases circulating round small loops, merging with other trains of bits, eventually arriving in stations called buffers.

These buffers can be 8 bits long (one byte) or 16 bits long (2 bytes) or even longer. The length is related to the architecture of the processor, and a 64-bit processor can handle addresses, integers and data path widths up to 64 bits, so effectively they naturally use numbers up (but not including) decimal 18,446,744,073,709,551,616! Computer people can’t read such long strings of bits of course so they convert the numbers to base sixteen (16) otherwise known as hexadecimal. It’s still very long, but can be handled and is less error prone than long strings of zeros and ones.

Round numbers are very useful as abbreviations. Saying nine thousand, eight hundred and seventy three is a lot more verbose than “about ten thousand” and is sufficiently accurate for many purposes.

One interesting use of round numbers is found in the nominal sizing of disk drives. To a computer person one byte is the smallest unit of storage. Bytes are usually grouped into ‘kilobytes’ where in this sense the prefix ‘kilo’ stands for one thousand and twenty four, and kilobytes are grouped into ‘megabytes’ where in this sense the prefix ‘mega’ stands for one thousand and twenty four again, and megabytes are grouped into ‘gigabytes’. This means that to a computer person a gigabyte contains 1,073,741,824 bytes. So this number (and numbers with the smaller prefixes of kilo and mega) are round numbers to computer people, because, if expressed in hexadecimal or binary bases these numbers end with long strings of zeros!)

There is a source of confusion here, because outside of the computer world, the prefixes of kilo, mega and giga are defined in terms of thousands. A kilogram is one thousand grams. Technically a megagram would be a thousand kilograms or a million of grams. This confusion impacts the computer world when computer disks size are given. To a computer disk manufacturer a gigabyte is one thousand million bytes, not a bit over one thousand and seventy three million bytes as mentioned above.

This leads to disappointment when purchasing disks. A nominally one hundred gigabyte disk will contain one hundred thousand thousand thousand bytes (100,000,000,000) but when when formatted will be able to contain less than ninety three gigabytes as the computer world counts bytes and that doesn’t take into account the overhead of the method of storing data on the disk. This overhead is necessitated by the need to hold the file names and locations on the disk itself so that the files can be retrieved.

There is no right or wrong way to consider bytes on disks and so computer people are in general now aware that if they buy a disk it will not seem (to them) to be quite as big as advertised. The moral is to ask what people mean when they use round numbers.

I was going to go into topics like giving change and Swedish rounding, but this post is already long enough. I will just mention that the topic of round numbers came to me because this is my fiftieth post! Fifty is sort of a round number, I suppose. It is halfway to a proper round number.