1. ARITHMETIC OPERATIONS IN OBSERVER'S MATHEMATICS

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1 1. ARITHMETIC OPERATIONS IN OBSERVER'S MATHEMATICS We cosider a ite well-ordered system of observers, where each observer sees the real umbers as the set of all iite decimal fractios. The observers are ordered by their level of depth, i.e. each observer has a depth umber (hece, we have the regular iteger orderig), such that a observer with depth k sees that a observer with depth < k sees ad deals (to be deed below) ot with a iite set of iite decimal fractios, but, actually, with a ite set of ite decimal fractios. We call this set W, i.e. it is the set of all decimal fractios, such that there are at most digits i the iteger part ad digits i the decimal part of the fractio. Visually, a elemet i W looks like _... _. _... _. Moreover, a observer with a give depth is uaware (or ca oly assume the existece) of observers with larger depth values ad for his purposes, he deals with iity. These observers are called aive, with the observer with the lowest depth umber the most aive. However, if there is a observer with a higher depth umber, he sees that a give observer actually deals with a ite set of ite decimal fractios, ad so o. Therefore, if we x a observer, the this observer sees the sets W 1,..., W k with 1 <... < k idicatig the depth level, ad realizes that the correspodig observers see ad deal with iity. Whe we talk about observers, we shall always have some xed observer (called `us') who oversees all others ad realizes that they are aive. The W -observer is the abbreviatio for somebody who deals with W while thikig that he deals with iity. We begi by deig sets W which cosist of all ite decimal fractios such that there are at most digits i the iteger part ad at most digits i the decimal part. That is, the set W cotais all elemets of the form a = a 0.a 1...a where the iteger part ca be writte as a 0 = b 1...b 0, where b 1,..., b 0, a 1,..., a {0, 1,..., 9}. Now, give c = c 0.c 1...c, d = d 0.d 1...d W we edow W with the followig arithmetic (+,,, ) - from W m - observer poit of view (m > ): Defiitio 1.1. Additio ad subtractio { c ± d, if c ± d W c ± d = ot deed, if c ± d / W where c ± d is the stadard additio ad subtractio, ad we write ((... (f 1 + f 2 )...) + f N ) = N f i for f 1,..., f N i the cotets of ay parethesis are i W, f 1,..., f N W. i=1 Defiitio 1.2. Multiplicatio c d = k=0 k m= c k k 1 m 1 where c, d 0, c 0 d 0 W, c k d m is the stadard product, ad k = m = 0 meas k 1 m 1 that c k = c 0 ad d m = d 0. If either c < 0 or d < 0, the we compute c d ad k 1 m 1 d m Khots, mathrelativity.com Page 1

2 dee c d = ± c d, where the sig ± is deed as usual. Note, if the cotet of at least oe paretheses (i previous formula) is ot i W, the c d is ot deed. Defiitio 1.3. Divisio c d = { r, if r W r d = c ot deed, if o such r exists Let = 2, so we are i W 2. Here are some examples of elemets of W 2 : 3.14, 99, 0.1 W 2 ad 0.115, 123.9, / W 2. Now, the examples of arithmetic: - we get 10 dieret r's = ( 2.08) = = ot deed = = ( 99.95) = ot deed = 88 ( 5) 2 19 = = ot deed = ( 2.64) = = ot deed = ot deed = = = {2, 2.01, 2.02, 2.04, 2.05, 2.06, 2.07, 2.08, 2.09} 1 3 = ot deed (sice o r exists). I case p > q, for W q -observer meas 10 q for W p -observer, ad 0 for W q -observer meas 10 q for W p -observer. Here we provide some basic examples to illustrate what might happe. 1. Additive associativity fails:(x + y) + z x + (y + z), e.g. let 10, 95, 35 W 2, the / W 2, hece ( ) + 2 ( 35) / W 2, but 10 + ( ( 35)) = 70 W 2 ; 2. Multiplicative associativity fails:(x y) z x (y z), e.g. let 50.12, 0.85, ad 0.61 W 2, the = ( ) ( ) = = 42.58, ad Khots, mathrelativity.com Page 2

3 ( ) = ( ) ( ) = = 25.65, whereas = ( ) ( ) = 0.48 ad ( ) = ( ) ( ) = = 24.04; 3. Distributivity fails: x (y + z) x y + x z, e.g. let 1.81, 0.74, 0.53 W 2, the = 1.27 ad ( ) = ( ) ( ) = = 2.24, whereas = ( ) ( ) = = 1.3 ad = ( ) ( ) = = 0.93, so that = 2.23; 4. Lack of the distributio law leads to the followig result: The statemet x y ad x z x (y + z) is false. Here x y r : x r = y. Assume that x y ad x z, what we wat to show is equivalet to showig that y + z x (r 1 + r 2 ) for some x, y, z, r 1 ad r 2. Let x = 0.17, r 1 = 0.85, r 1 = 0.63, y = = 0.( ) ( ) = 0.08 ad z = = 0.( ) ( ) = The y + z = 0.14, but r 1 +r 2 = 1.48 ad = ( ) ( ) = = I fact, x is ot divizor of y + z. This is because if we let = ( ) 0.9 = 0.09 < 0.14 ad = ( ) ( ) = 0.09 < 0.14, but = 0.17 > Multiplicative iverses do ot ecessarily exist, or if they do, they are ot ecessarily uique i W. Here are some examples: let 2 W, the 0.5 W 2 is the uique iverse of 2 for ay W. O the other had, 3 will ot have a iverse i ay W. Now, let 2 1 = 0.5, the (0.5) 1 is actually the followig set {2, 2.01, 2.02, 2.03, 2.04, 2.05, 2.06, 2.07, 2.08, 2.09} W 2. Therefore, (2 1 ) 1 is ot ecessarily 2, hece all we ca claim is that if x 1 exists, the x { (x 1 ) 1}. Further, if a iverse of a elemet exists i W, it does ot ecessarily exist i W m for m, idepedet of the order of mad, e.g. if 0.91 W 2, the (0.91) 1 = {1.1, 1.11, 1.12, 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19} W 2, but (0.91) 1 / W 4, o the other had, 16 1 = W 4, but 16 1 / W Square roots do ot ecessarily exist. Some examples are, if 4 W, the 4 = 2 for ay ad 3 does ot exist i = 2. To show that, cosider = ( ) ( ) = = 2.99 ad = ( ) ( ) = = Further, if a square root of a elemet exists i W, it does ot ecessarily exist i W m for m, idepedet of the order of m ad, e.g. 2 = 1.42 W2, sice = ( ) ( ) = = 2, but 2 / V 4, sice = ( ) ( ) = ad = Also, 1.01 = W 4, sice = ( ) ( ) = = 1.01, but 1.01 / W 2, sice = 1 ad = ( ) ( ) = = Next, some basic theorems ca be stated for W : Khots, mathrelativity.com Page 3

4 . 1. Ay W has zero divisors: = 0; If p W with p 2, 5 a prime i the usual sese, the p 1 / W for ay W ; 3. x, y W with x, y 0, x y W. 4. If x, y, t, u W ad x t 0 ad y u 0 ad x y W, the t u W ad x y t u 5. If give a W such that there is a uique a 1 W, the a 1; 6. If a < 1 ad a 1 exists, the {a 1 } > 1; 7. If {a 1 } > 1, the a < 1. Let's cosider ow additioal valuable properties of itroduced arithmetic. 1. Stadard multiplicatios idetities become wrog, for example We have (x + y) 2 x (xy) + y 2 Theorem 1.4. P ((a + b) (a + b) = (a a + 2 (a b)) + b b) < 1, where P is a probability. We ca see a proof below. Let = 2. The 1. Left side of equality is ( ) 2 ( ) = = 13.99, right side cosists from two parts. First, = 1.73; secod, 2 2 ( ) = 6.38, ad ally = That meas ) = I.e left side equals to right. But ow let's cosider the followig calculatios. 2. Left side of equality is ( ) 2 ( ) = = 16.89, right side cosists from two parts. First, = 1.73; secod, 2 2 ( ) = 7.28, ad ally = That meas = I.e left side does ot equal to right. Let's cosider ow a radom variable δ 1 = (a + b) (a + b) ((a a + 2 (a b)) + (b b)) where a, b 0, ad δ 1, ad all elemets of right side belog to W. Let's = 2. Usig direct calculatio we ca build F 1 (x) - distributio fuctio of δ 1, where F 1 (x) = P (δ 1 < x), P is a probability. Graph of F 1 (x) you ca see o Fig. 1. Khots, mathrelativity.com Page 4

5 F X Figure 1. Graph F 1. Geeral proof for W you ca see below. b) (a + b) W, the δ 1 = 0. let's cosider ow a = 0. 9 }.{{.. 9} ad (a+ b) (a+ b) = a+ b = 1. } 0.{{.. 07} b b = 0, ad 2 (a b) = 0. I.e. δ We have also the followig theorem. Theorem 1.5. If a, b o-egative itegers i W ad (a + ad b = 0. 0 }.{{.. 08}. The = , but a a < 1, P (c (a + b) = c a + c b) < 1, where P is a probability. Below you ca see a proof. Let's = 2. The } {{ } 1. Left side of equality is 2 2 ( ) = = 18, right side cosists from two parts. First, = 6, the = 12 ad = 18 I.e. left side equals to right. But go to ext calculatios. 2. Left side of equality is ( ) = = 8.95, right side cosists from two parts. First, = 7.55, the = 1.36 è = I.e. left side does ot equal to right. Let's cosider a radom variable δ 2 = c (a + b) (c a + c b) Khots, mathrelativity.com Page 5

6 F X Figure 2. Graph F 2. where a, b, c 0, ad δ 2 ad all elemets of right side belog to W. Let's = 2. Usig direct calculatios we ca build F 2 (x) - distributio fuctio of δ 2, where F 2 (x) = P (δ 2 < x), where P is a probability. Graph of F 2 (x) you ca see o Fig. 2. Geeral proof for W you ca see below. If a, b, c - o-egative itegers i W ad a (b c) W, the δ 2 = 0. Let's cosider ow a = 2, b = 0. 9 }.{{.. 9} ad c = 0. 0 }.{{.. 01}. The b c = 0, a (b c) = 0, a b = , ad (a b) c = I.e. δ Theorem 1.6. Let's δ 3 = c (a b) (c a) b. The 0 < P (δ 3 = 0) < 1, where P is a probability. You ca see a proof of this theorem below. Let's = 2. The 1. Left side of this equality is 2 2 (3 2 6) = = 36, right side cosists from two parts. First, = 6, the = 36. I.e left side equals to right. But let's cosider the followig calculatios. 2. Left side is ( ) = = 4.27, for right side we get rst, = 7.55, the = Ad left side does ot equal to right Let's cosider a radom variable δ 3 = c (a b) (c a) b Khots, mathrelativity.com Page 6

7 Figure 3. Graph F 3., where a, b, c 0, ad δ 3 ad all elemets of right side belog to W. If we take = 2, the usig direct calculatios we ca build F 3 (x) - distributio fuctio of δ 3, where F 3 (x) = P (δ 3 < x), ad P is a probability. Graph of F 3 (x) you ca see o Fig. 3. Geeral proof for W you ca see below. If a, b, c are o-egative itegers i W ad c (a b) W, the δ 3 = 0. Let's cosider c = 2, a = è b = 0. 0 }.{{.. 01}. The δ 3 = 2 ( ) ( ) = = = Khots, mathrelativity.com Page 7