Power of two

In Mathematical, a power of two is any positive Entier put in exhibitor of the number Two; in other words, twice itself a certain number of times. Let us note that is a power (power zero) of two. Written in binary, a power of two is always of the form 10000… 0, like a power of ten in the system Décimal.

As two is the base of the binary system, the powers of two are important in Informatique. In a more precise way, two with the power N is the number of ways in which the bits in a binary entirety length N can be arranged, and thus these numbers represent a power of two minus one indicate the higher limit of the whole if they are interpreted as a whole without sign (to represent the power of 2 itself (without the -1) would ask for a bit of more).

The numbers of this form appear sometimes in the software. For example, in the Video game the legend of Zelda for the game console Nintendo 8 bits, one can only work with long numbers of 255 bits at the same time - the result of a Octet, which is length 8 bits, used to store a number, given by a maximum value of 2⁸-1 = 255.

The memory sizes of the computers use also the powers of 2. A Octet contains (2³) bits, and a Kilo-octet contains 1.024 (2¹⁰) bytes. Almost all the registers of processors have sizes which are powers of two (one is currently starting a transition from 32 to 64 on PC).

The powers of two occupied a place in old the disk players: cut sector, number of sector per track, and/or number of track by surface were often powers of two. The discs then had what one named a geometry (fixed number of sectors per track), which ended up being abandoned with the profit of one recording density about constant (more sectors towards outside) at the price of a little additional electronics.

The size of the logical building block remained, for historical reasons and of convenience of calculation, a power of two.

One finds in the video resolutions of the numbers which are not powers of two, but are often the sum or the product of two or three powers of two, or the powers of two minus one. For example, 640 = 512 + 128, and 480 = 32 × 15, the resolution of the screens 14′ ′.

One names Prime number of Mersenne a Prime number of the form $2^N-1 \,$ . For example, the prime number 31 and which is written in the form 32 (2⁵) less 1.

Quarantes first powers of two

Powers of two which have as exhibitors of the powers of two

The modern storage cells and the registers often handle a number of bits which is a power of two, the most frequent powers which appear are those whose exhibitor is also a power of two. - a short list of some of enters:

Other powers of two remarkable

2²⁴ = 16.777.216 - The number of single colors which can be posted in true Couleurs, which is used by the majority of the screens of computers.

This number is the result of the use of the system with three channels RVB, with 8 bits for each channel, or 24 bits on the whole.

Theorems

the sum of the powers of 2 is equal to the following power of two less un.

S_n = \ sum_ {i=0} ^n2^i = 2^ {n+1} - 1

a power of two is equal to the sum of all the lower powers of two more un.

2^ {n+1} = S_n + 1 \,

2^n = S_ {n-1} + 1 \,

an entirety is divisible by $\ 2^n$ if and only if the number made up of its $\ N$ last figures is divisible by $\ 2^n$ .

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