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{\displaystyle b} , , {\displaystyle b} , . .[1] {\displaystyle r} , {\displaystyle m=r\cdot b} , {\displaystyle b^{r}=b^{\frac {m}{b}}} . {\displaystyle b} , {\displaystyle b} e = 2,718281828[2]. {\displaystyle b} {\displaystyle b=3} . , ( ), ( ) .

. , ( , , ), [3] .[1] . .

, . n - b {\displaystyle n{\tfrac {\ln b}{b}}} ( ). ( , ) b {\displaystyle {\tfrac {\ln b}{b}}} , b = e, b b = 3.

[ | -]

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{\displaystyle b} -

{\displaystyle a_{n-1}\;a_{n-2}\ldots \;a_{1}\;a_{0},}

:

{\displaystyle \sum _{k=0}^{n-1}a_{k}\cdot b^{k}}

, :

{\displaystyle a_{n-1}\cdot b^{n-1}+a_{n-2}\cdot b^{n-2}+\ldots +a_{1}\cdot b^{1}+a_{0}\cdot b^{0},}

, , :

{\displaystyle ((\ldots (a_{n-1}\cdot b+a_{n-2})\cdot b+a_{n-3})\ldots)\cdot b+a_{0}.}

:

1011002 =

= 1 25 + 0 24 + 1 23 + 1 22 + 0 21 + 0 20 =

= 1 32 + 0 16 + 1 8 + 1 4 + 0 2 + 0 1 =

= 32 + 8 + 4 + 0 = 4410

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1. , .

2. . , .

1. , . . , 0.

2. , .

{\displaystyle 44_{10}} :

44 2. 22, 0

22 2. 11, 0

11 2. 5, 1

5 2. 2, 1

2 2. 1, 0

1 2. 0, 1

, . {\displaystyle 101100_{2}}

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.

, 2 (2 , , , (2³=8), 3, ). :

000 0 100 4

001 1 101 5

010 2 110 6

011 3 111 7

, 2 (2 , , , (24=16), 4, ). :

0000 0 0100 4 1000 8 1100 C

0001 1 0101 5 1001 9 1101 D

0010 2 0110 6 1010 A 1110 E

0011 3 0111 7 1011 B 1111 F

:

1011002

101 100 → 548

0010 1100 → 2C16

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-.

0 000 4 100

1 001 5 101

2 010 6 110

3 011 7 111

0 0000 4 0100 8 1000 C 1100

1 0001 5 0101 9 1001 D 1101

2 0010 6 0110 A 1010 E 1110

3 0011 7 0111 B 1011 F 1111

:

548 → 101 100

2C16 → 0010 1100

8- 16- [ | -]

8 16 , , , , . , 1100,0112 14,38 C,616.

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1100,0112 . 12 (. ), :

{\displaystyle 0,011=0\cdot 2^{-1}+1\cdot 2^{-2}+1\cdot 2^{-3}=0+0,25+0,125=0,375.}

, 1100,0112 = 12,37510.

, 2 .

, , .

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, . , , () . , . 103,62510 .

, , 10310 = 11001112.

0,625 2. 0,250. 1.

0,250 2. 0,500. 0.

0,500 2. 0,000. 1.

, 1012. 103,62510 = 1100111,1012

.

, , , , , . . , , , 0,626.

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[ | -]

{\displaystyle x} {\displaystyle b} - ( , ) {\displaystyle b} :

{\displaystyle x=(a_{n-1}a_{n-2}\ldots a_{1}a_{0},c_{1}c_{2}\ldots)_{b}=\sum _{k=0}^{n-1}a_{k}b^{k}+\sum _{k=1}^{\infty }c_{k}b^{-k}}

{\displaystyle a_{k}} ( ), {\displaystyle c_{k}} ( ), {\displaystyle n} .

{\displaystyle b} - , {\displaystyle {\frac {q}{b^{m}}}} , {\displaystyle m} {\displaystyle q} :

{\displaystyle {\frac {q}{b^{m}}}=(a_{n-1}a_{n-2}\ldots a_{1}a_{0},c_{1}c_{2}\ldots c_{-m})_{b}=\sum _{k=0}^{n-1}a_{k}b^{k}+\sum _{k=1}^{m}c_{k}b^{-k},}

{\displaystyle (a_{n-1}a_{n-2}\ldots a_{1}a_{0})_{b}} {\displaystyle (c_{1}c_{2}\ldots c_{-m})_{b}} {\displaystyle b} - {\displaystyle q} {\displaystyle b^{m}} .

, {\displaystyle {\frac {q}{b^{m}}}} , .

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(, ) , {\displaystyle (0,1,\ldots,b-1)} , {\displaystyle \left(0-\left({\frac {b-1}{2}}\right),1-\left({\frac {b-1}{2}}\right),\ldots,(b-1)-\left({\frac {b-1}{2}}\right)\right)} . , , {\displaystyle b} . .[4] , , , .

{\displaystyle (-1,0,1)} . .

[ | -]

: -

, -:

-2 -

-3 -

-10 -

[ | -]

: , , .

:

b = ⅓ , [ 2463 ],

b = ½ [ 2463 ],

b = φ = 1,61 .[5]

[ | -]

[6][7] . , , .

, , 0 1.

{\displaystyle \langle \rho,A\rangle } , {\displaystyle \rho } , A . , A :

{\displaystyle B_{R}=\{0,1,2,\dots,R-1\},}

{\displaystyle D_{R}=\{-r_{1},-r_{1}+1,\dots,-1,0,1,\dots,r_{2}-1,r_{2}\},} {\displaystyle r_{1},r_{2}\geq 0} {\displaystyle R=r_{1}+r_{2}+1} . {\displaystyle r_{1}=0} {\displaystyle D_{R}} {\displaystyle B_{R}} .

( j ):

{\displaystyle \langle \rho =j{\sqrt {R}},B_{R}\rangle.} [7]

: {\displaystyle \langle \rho =\pm j{\sqrt {2}},\{0,1\}\rangle;}

{\displaystyle \langle \rho ={\sqrt {2}}e^{\pm j\pi /2},B_{2}\rangle.} [6]

: {\displaystyle \langle \rho =-1\pm j,\{0,1\}\rangle;}

{\displaystyle \langle \rho =2e^{j\pi /3},\{0,1,e^{2j\pi /3},e^{-2j\pi /3}\}\rangle;} [8]

{\displaystyle \langle \rho ={\sqrt {R}},B_{R}\rangle,} {\displaystyle \varphi =\pm \arccos {(-\beta /2{\sqrt {R}})}} , {\displaystyle \beta <\min\{R,2{\sqrt {R}}\}} , R;[9]

{\displaystyle \langle \rho =-R,A_{R}^{2}\rangle,} {\displaystyle A_{R}^{2}} {\displaystyle r_{m}=\alpha _{m}^{1}+j\alpha _{m}^{2}} , {\displaystyle \alpha _{m}\in B_{R}.} : {\displaystyle \langle -2,\{0,1,j,1+j\}\rangle;} [8]

{\displaystyle \langle \rho =\rho _{2}^{},\{0,1\}\rangle,} {\displaystyle \rho _{2}={\begin{cases}(-2)^{1/2}&{\mbox{if}}\ m\ {\mbox{even}},\\j(-2)^{(m-1)/2m}&{\mbox{if}}\ m\ {\mbox{odd}}.\end{cases}}} .[10]

 

2, −2 −1 :

{\displaystyle \rho =2} : {\displaystyle 2=(10)_{\rho }} ( );

{\displaystyle \rho =-2} : {\displaystyle 2=(110)_{\rho }} , {\displaystyle -2=(10)_{\rho }} , {\displaystyle -1=11_{\rho }} (- );

{\displaystyle \rho =-\rho _{2}} : {\displaystyle 2=(10100)_{\rho }} , {\displaystyle -2=(100)_{\rho }} , {\displaystyle -1=101_{\rho }} ( );

{\displaystyle \rho =j{\sqrt {2}}} : {\displaystyle 2=(10100)_{\rho }} , {\displaystyle -2=(100)_{\rho }} , {\displaystyle -1=(101)_{\rho }} ( );

{\displaystyle \rho =-1+j} : {\displaystyle 2=(1100)_{\rho }} , {\displaystyle -2=(11100)_{\rho }} , {\displaystyle -1=(11101)_{\rho }} ( );

{\displaystyle \rho ={\frac {-1+j{\sqrt {2}}}{2}}} : {\displaystyle 2=(1010)_{\rho }} , {\displaystyle -2=(110)_{\rho }} , {\displaystyle -1=(111)_{\rho }} ( ).

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. . ,

{\displaystyle {\operatorname {hyper4} (a,b)=\operatorname {hyper} (a,4,b)=a^{(4)}b=a\uparrow \uparrow b= \atop {\ }}\quad {\underbrace {a^{a^{\cdot ^{\cdot ^{a}}}}} \atop {b{\mbox{ times}}}}\quad {=a\to b\to 2 \atop {\ }}}

́ .

 

( ) () , . . IEEE 754. , .

[ | -]

, , (. Decimal separator (.)) , (floating point (.)). , , .

[ | -]

, ( , , , ) . . , .

( ) , . , , 6 2 , 123 456,78. , 8 1,2345678; 1 234 567,8; 0,000012345678; 12 345 678 000 000 000 , 10 0 16, 8+2=10.

, , FLOPS ( . floating-point operations per second [] ), .

[ | -]

:

( ),

( ),

,

( , ).

[ | -]

, ( ) [0 1), 0 ⩽ a < 1. , , .[ 2698 ]

: (, 0,0001 0,000001×102, 0,00001×101, 0,0001×100, 0,001×10−1, 0,01×10−2, ), ( ) , 1 () 10 ( ), 1 () 2 ( ), 1 ⩽ a < 10. ( 0) . , 0, () 0.

( ) ( 0) 1 ( ), , IEEE 754. , 2 ( , .), .

(, ), , m be (m ; b , 10; e ), , E ( . exponent). 10&. , 1,528535047×10−25 1.528535047E-25.

[ | -]

, :

(radix point) .

, (, , ) .

. , 8 ( 4- 5- ). , 00012345 1,2345 ( ).

() . () , {\displaystyle aq^{n}} , {\displaystyle a} , , , {\displaystyle 1\leq a<q} , {\displaystyle n} , {\displaystyle q} , ( 10). ( ) , , . , ( ) , 152 853,5047 , 1,528535047×105 . , 0.

, . , . ́, 1528535047 5. , 105 1,528535047, 5 . , , 2, 10. , , , , 0 . , 9 +10010000 +00100. , , (0,1), 1/3, .

, . , (3 ) ,

0,12 × 0,12 = 0,0144

(1,2010−1) × (1,2010−1) = (1,4410−2).

0,120 × 0,120 = 0,014.

, .





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: 2016-11-24; !; : 1459 |


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