Binary vs. Decimal Measurements

Binary vs. Decimal Measurements

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One of the more difficult problems you will face when working with computer hardware, especially hard drives, is the two different measurement definitions or terms used to calculate drive capacity. Capacity measurements are usually expressed in kilobytes (thousands of bytes), in megabytes (millions of bytes), or gigabytes (billions of bytes), however, due to a mathematical coincidence there are two different meanings for each of these measures.

Computers are digital, and with that store data using binary numbers, or powers of two, although we are accustomed to using decimal numbers, expressed as powers of ten. As it turns out, two to the tenth power, 2^10, is 1,024, which is very close in value to 1,000 (10^3).  Similarly, 2^20 is 1,048,576, which is approximately 1,000,000 (10^6), and 2^30 is 1,073,741,824, close to 1,000,000,000 (10^9). As computer development became more prominent and binary numbers began to be used on a regular basis, computer scientists took note of this similarity and began using the abbreviations normally associated with decimal numbers, and applied them to binary numbers. This led to 2^10 being given the prefix “kilo”, 2^20 given the prefix “mega”, and 2^30 referred to as “giga”.

This shorthand reference works well when used between technicians who regularly work in computer development, as they know what they are referring to, (and no one else really cares). However, when computers entered the mainstream, the dual reference began leading to quite a bit of confusion and inconsistency. In many areas of development, only binary values are used. As an example, 64 MB of RAM memory always means 64 times 1,048,576 bytes, never 64,000,000. Lending confusion to this mess though, in some areas only decimal values are used such as when the term, “56K modem” works at a maximum speed of 56,000 bits per second, not 57,344.

It’s no secret that storage devices are the single largest area of confusion in this regard, as some drive manufacturers use the decimal method, while others use the binary method when advertising their drive capacities. Even some software companies play this game, with some software packages using binary megabytes and gigabytes, and others using decimal megabytes and gigabytes. If you are using smaller numbers, the difference between decimal and binary is rather insignificant, but as the numbers grow larger so does the disparity. As an example, there is only a 2.4% difference between a decimal and a binary kilobyte, but when you calculate a megabyte, this difference increases to approximately 5%. A gigabyte produces a difference of approximately 7.5%, which is a rather significant difference. Not too many years ago this difference, as it pertains to hard drives that is, wasn’t all that noticeable when drive manufacturers advertised the size of their drives in decimal format, and you partitioned and formatted your drive in binary. Today, however, people are beginning to notice the difference between the two measures. As an example, should you purchase an “80 GB” hard drive, in all probability it will partition and format to about 74.2 to 76.3 gigabytes. Don’t worry, there’s nothing wrong with the drive, it’s just that the manufacturer stated the 80 GB in decimal format, but Windows (MS-DOS) partitioned and formatted the disk in binary gigabytes. There are a few other issues with large hard drive capacities, but we’ll address those in the drive limitations and barriers section of this topic area.

Another issue of importance is that of converting between binary gigabytes and binary megabytes. Decimal gigabytes and megabytes differ by a factor of 1,000, however binary measurement differs by 1,024. So the same 80 GB hard disk is 80,000 MB in decimal terms, however the 76.3 binary gigabytes are equal to 78,131 binary megabytes (76.3 times 1,024).

Figure 1:

This is an example of a Windows 9x display of the capacity of an 8 GB hard disk drive. Take note of the disparity between the number of bytes and and GB values, which are obviously in binary format.

Although a bit off the subject of hard drives, but relevant nonetheless, is the issue of the mathematical differences in definitions of “mega” or “giga”. A classic example would be the PCI bus, which has a theoretical maximum bandwidth of 133.3 Mbytes/second as the result of the fact that it is 4 bytes wide and runs at 33.3 MHz. Unfortunately though, the “M” in “MHz” is 1,000,000, but the “M” in “Mbytes/second” is 1,048,576. Therefore, the bandwidth of the PCI bus is more properly stated as 127.2 Mbytes/second (4 times 33,333,333 divided by 1,048,576).

Even with all of this confusion, there appears to be some light at the end of the tunnel. The Institute for Electrical and Electronics Engineers (IEEE) has proposed an entirely new naming convention for binary numbers in an effort to eliminate some of this confusion. Under there proposal, when binary numbers are in use, the third and fourth letters in the prefix are changed to “bi”. As an example, “mega” becomes “mebi”. One megabyte would be 10^6 bytes, however one mebibyte would be 2^20 bytes and the abbreviation would become “1 MiB” instead of “1 MB”. Below you will find a table showing the decimal and binary values and their conversion values. While “bytes” are shown in this table, the prefix could apply to any unit of measure.

Decimal Name	Decimal Abbr.	Decimal Power	Decimal Value	Binary Name	Binary Abbr.	Binary Power	Binary Value
Kilobyte	kB	10^3	1,000	Kibibyte	kiB	2^10	1,024
Megabyte	MB	10^6	1,000,000	Mebibyte	MiB	2^20	1,048,576
Gigabyte	GB	10^9	1,000,000,000	Gibibyte	GiB	2^30	1,073,741,824
Terabyte	TB	10^12	1,000,000,000,000	Tebibyte	TiB	2^40	1,099,511,627,776

While it appears that no one is jumping on the bandwagon to accept this new standard, should you want to research this issue a little further, click here: International System of Units (SI) – Definitions of the SI Units: The Binary Prefixes. We have decided to wait and allow the industry as a whole make the switch before we decide to do so, in order to avoid even more confusion.

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