## Power of two

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Either way, one less than a power of two is often the upper bound of an integer in binary computers. As a consequence, numbers of this form show up frequently in computer software.

Powers of two are often used to measure computer memory. A byte is now considered eight bits an octet , resulting in the possibility of values 2 8. The term byte once meant and in some cases, still means a collection of bits , typically of 5 to 32 bits, rather than only an 8-bit unit. The prefix kilo , in conjunction with byte , may be, and has traditionally been, used, to mean 1, 2 However, in general, the term kilo has been used in the International System of Units to mean 1, 10 3.

Nearly all processor registers have sizes that are powers of two, 32 or 64 being most common. Powers of two occur in a range of other places as well. For many disk drives , at least one of the sector size, number of sectors per track, and number of tracks per surface is a power of two.

The logical block size is almost always a power of two. Numbers that are not powers of two occur in a number of situations, such as video resolutions, but they are often the sum or product of only two or three powers of two, or powers of two minus one. Put another way, they have fairly regular bit patterns.

A prime number that is one less than a power of two is called a Mersenne prime. For example, the prime number 31 is a Mersenne prime because it is 1 less than 32 2 5. Similarly, a prime number like that is one more than a positive power of two is called a Fermat prime —the exponent itself is a power of two.

A fraction that has a power of two as its denominator is called a dyadic rational. The numbers that can be represented as sums of consecutive positive integers are called polite numbers ; they are exactly the numbers that are not powers of two. The geometric progression 1, 2, 4, 8, 16, 32, … or, in the binary numeral system , 1, 10, , , , , … is important in number theory.

The sum 31 multiplied by 16 the 5th term in the series equals , which is a perfect number. This is a restatement of our formula for geometric series from above. Applying this to the geometric progression 31, 62, , , which results from 1, 2, 4, 8, 16 by multiplying all terms by 31 , we see that 62 minus 31 is to 31 as minus 31 is to the sum of 31, 62, , Therefore, the numbers 1, 2, 4, 8, 16, 31, 62, and add up to and further these are all the numbers that divide For suppose that p divides and it is not amongst these numbers.

Now p cannot divide 16 or it would be amongst the numbers 1, 2, 4, 8 or Therefore, 31 cannot divide q. And since 31 does not divide q and q measures , the fundamental theorem of arithmetic implies that q must divide 16 and be amongst the numbers 1, 2, 4, 8 or Let q be 4, then p must be , which is impossible since by hypothesis p is not amongst the numbers 1, 2, 4, 8, 16, 31, 62, or These patterns are generally true of any power, with respect to any base. Because data specifically integers and the addresses of data are stored using the same hardware, and the data is stored in one or more octets 2 3 , double exponentials of two are common.

Has this fearful pestilence no power to restrain the appetites and passions of the people? But I have a secret dread of the character and power of Alcibiades. If a servant complained of being abused, his master had no power to retain him. Then I shall have to put it out of your power to carry out your threat. He was forced to admit that the girl still had power to trouble him. Meaning "one who has power" is late 14c.

Meaning "specific ability or capacity" is from early 15c. Meaning "a state or nation with regard to international authority or influence" [OED] is from Used for "a large number of" from s.

Meaning "energy available for work is from Sense of "electrical supply" is from Phrase the powers that be is from Rom. As a statement wishing good luck, more power to someone is recorded from A power play in ice hockey so called by Power failure is from ; power steering from Earlier it meant "make powerful" s. In physics , the amount of energy put out or produced in a given amount of time.

Power is often measured in watts or kilowatts. In mathematics , a power is a number multiplied by itself the number of times signified by an exponent placed to the right and above it. In addition to the idioms beginning with power. Synonyms Examples Word Origin. Atomic energy powers the new submarines.

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