# FRACTION CALCULATOR Figure 1. Mergers and Acquisitions achieving the expected cost savings (Source: McKinsey 2004 <20> ).

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Successful M & As, those that are in the right part of Figure 2, are sometimes described by the expression 1 + 1 = 3 (see, for example, Beechler, 2003 <21> ) because this expression reflects the fact that after a M và A the sum of the two combined entities, 3, is bigger than the two parts considered separately, 1 & 1.

In a similar way, the Cambridge Business English Dictionary (2011) <22> de- fines that when two companies or organizations join together, they achieve more và are more successful than if they work separately, or in other words, the merger results in 2 + 2 = 5. Therefore, synergy emerges when the cooperation of two systems gives a result greater than the sum of their individual components.

Most studies of M và As (see for example (Cartwright & Schoenburg, 2006 <23> ; Krug và Aguilera, 2005 <24> ; Agraval và Jaffe, 2000 <25> ) find that the majority of M & As do not meet their expectations, see the left part of Figure 2. The main reasons for the failure of many M & A are

1) Poor cultural fit or lack of cultural compatibility,

2) Overestimation of synergy effects,

3) Underestimation of cost involved in the M và A process,

4) Employee turnover.

When M & As bởi vì not meet their expectations, it is possible to argue that these situations are correctly reflected by expressions 1 + 1 = 1, or even 1 + 1 = 0. This state of affairs reflects negative synergy or system friction.

Expression 1 + 1 = 3 may also describe synergies in other areas. Michael Angier (2005) <26> defines synergy as the phenomenon of two or more people getting along & benefiting one another, i.e., the combination of energies, resources, talents và efforts equal more than the sum of the parts. It is possible to describe this phenomenon by the expression 1 + 1 = 3.

In addition, the expression 1 + 1 = 3 also emerges in other areas, for example in the exploration of the features of visual information (Tufte, 1990 <27> ). For example in an expression of two words, e.g., “every thing”, the white space between the words provides additional information. Its absence changes the meaning, e.g., without the white space, we have “everything”. Adding two words (symbols), i.e., 1 + 1, we obtain 3 meaningful symbols. It is possible to gọi these expressions 1 + 1 = 3 and 2 + 2 = 5, synergy arithmetic.

Furthermore, there are examples of non-Diophantine arithmetic in everyday life. Imagine that you come khổng lồ a supermarket và you can see an advertisement “Buy one, get one free”. It actually means that you can buy two items for the price of one. Such advertisement may refer khổng lồ almost any product: bread, milk, juice etc. For example, if one bottle of orange juice costs \$2, and you get two for one, then we have the equality 2 + 2 = 2. This is incorrect in the conventional arithmetic but is true for some non-Diophantine arithmetic, as we will show below.

Another example: when a cup of milk is added lớn a cup of popcorn then only one cup of mixture will result because the cup of popcorn will very nearly absorb a whole cup of milk without spillage. So we have 1 + 1 = 1. This is impossible to replicate with conventional arithmetic but it is true for some non-Diophantine arithmetics.

One more example is when you want khổng lồ buy a car, which according to the newspaper advertisement, costs \$19,990. Coming lớn the dealership, you find that the price is five dollars more. Bởi you think that the new price is different from the initial one or you consider it practically one và the same price? It is natural khổng lồ suppose that any sound person has the second opinion. Consequently, we come to the same “paradox”: if k is the price of the car, then in the Diophantine arithmetic k + 5 is not equal lớn k, while with respect lớn you, they are basically identical.

Critics may object that we use Non-Diophantine arithmetics to explain these phenomena. They may say that we use the conventional arithmetic but only transform its operations according lớn some formulas. This objection is similar to lớn the 18th century Europe claim that people bởi not use negative numbers but only employ positive numbers with additional symbols.

Now let us look whether the laws of non-Diophantine arithmetic can reflect the economic và psychological phenomena considered above.

4. Laws of Non-Diophantine Arithmetics

Here we consider only arithmetics in the mix W of whole numbers. We start with a more general concept of a prearithmetic (Burgin, 2010 <23> ).

If X is a subset of the phối R of all real numbers, then the arithmetical completion of X consists of all sums and products of elements from X. For instance, if we take the phối 1 that has only one element 1, then its arithmetical completion is the set N = 1 , 2 , 3 , ⋯ of all natural numbers because in R, any natural number is the sum of some quantity of the number 1.

Let us consider two functions f: W ® R and g: R ® W. Functions f & g allow defining two new operations in the mix W:

a ⊕ b = g ( f ( a ) + f ( b ) )

a ∘ b = g ( f ( a ) ⋅ f ( b ) ) .

Here a và b are whole numbers, + is addition & × is multiplication of real numbers, while ⊕ is addition and ∘ is multiplication of numbers in prearithmetic defined by functions f và g. Let us take the mix A which is the domain name of f, i.e., the subset of W where f is defined.

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The structure A = 〈 A , ⊕ , ∘ 〉 is called a whole-number prearithmetic. In other words, a whole number prearithmetic is a set of whole number with operations ⊕ và ∘ .

Naturally the conventional arithmetic W is a whole-number prearithmetics. Another example of whole-number prearithmetics is a modular arithmetic, which is studied in mathematics và used in physics và computing. In modular arithmetic, operations of addition và multiplication are defined but in contrast khổng lồ the conventional arithmetic, its numbers “wrap around” upon reaching a certain value, which called the modulus. For instance, when the modulus is equal to lớn 10, the modular arithmetic Z10 contains only ten numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 & when the result of the operation the conventional arithmetic is larger than 10, then it is reduced to these numbers in the modular arithmetic. Readers can find information about modular arithmetics in many books và on the Internet. Here we only give some examples for the modular arithmetic Z10:

2 + 2 = 4

but

5 + 5 = 0

and

7 + 8 = 5

3 × 3 = 9

but

5 × 5 = 5

and

4 × 4 = 6

Note that in a general case, operations ⊕ và ∘ are partial, i.e., they are not defined for all numbers from W.

When a whole-number prearithmetic satisfies the following conditions, it becomes a general Non-Diophantine arithmetic.

Condition A1. F: W ® R is a total function.

Condition A2. The arithmetical completion of the image of f is a subset of the tên miền of g.

Condition A3. The composition g ∘ f : W ® W is a projection, i.e., its image coincides with the whole W.

For instance, condition A2 is true when g is also a total function, e.g., g ( x ) = ⌈ x ⌉ , or when f maps a whole number into a whole number và g is defined for all whole numbers, e.g., f ( x ) = 2 x , và g ( x ) = 3 x .

An important class of Non-Diophantine arithmetics is formed by projective arithmetics. Khổng lồ build a projective arithmetic, we take a non-decreasing function h: R ® R & its inverse relation h−1 defining the following two functions hT & hT:

h T ( x ) = ⌈ h ( x ) ⌉

h T ( x ) = ⌊ h − 1 ( x ) ⌋

Here ⌈ a ⌉ is the ceiling of a natural number a, which is the least whole number that is larger than a, & ⌊ a ⌋ is the floor of a natural number a, which is the largest whole number that is less than a.

For instance, taking the number 2.75, we have ⌈ 2.75 ⌉ = 3 & ⌊ 2.75 ⌋ = 2 .

Taking f = h T & g = h T , we build the whole-number prearithmetic A W = 〈 W , ⊕ , ∘ 〉 with

a ⊕ b = h T ( h T ( a ) + h T ( b ) )

a ∘ b = h T ( h T ( a ) ⋅ h T ( b ) )

This prearithmetic is a projective whole-number arithmetic if the following conditions are satisfied:

1) h T ( 0 μ ) = 0 ;

2) h ( x ) is a strictly increasing function;

3) for any elements a và b from U from a ≤ b , we have h T ( S a ) − h T ( a ) ≤ h T ( S b ) − h T ( b ) .

Here Sa is the number that follows a in the Diophantine arithmetic, e.g., S2 = 3 and S7 = 8.

It is possible to find a theory of this & other Non-Diophantine arithmetics in Burgin, 1977; 1997; 2007; và 2010. Here we consider only simple examples & some properties of Non-Diophantine arithmetics because the goal of this work is a demonstration of a possibility of the rigorous mathematics to correctly và consistently interpret seemingly paradoxical statements, which describe situations in various spheres of real life.

Example 1. Let us take the function f ( x ) = 2 x and its inverse function f − 1 ( x ) = ( 1 / 2 ) x . In this case, f T ( a ) = f ( a ) for all whole numbers a & if c = 2d, then f T ( c ) = f − 1 ( c ) . This allows us to lớn build the whole-number arithmetic A = 〈 W , ⊕ , ∘ 〉 and perform summation and multiplication in it finding some sums và products:

1 ⊕ 1 = ( 1 / 2 ) ( 2 ⋅ 1 + 2 ⋅ 1 ) = ( 1 / 2 ) ( 2 + 2 ) = ( 1 / 2 ) ( 4 ) = 2

2 ⊕ 2 = ( 1 / 2 ) ( 2 ⋅ 2 + 2 ⋅ 2 ) = ( 1 / 2 ) ( 4 + 4 ) = ( 1 / 2 ) ( 8 ) = 4

However,

1 ∘ 1 = ( 1 / 2 ) ( ( 2 ⋅ 1 ) ⋅ ( 2 ⋅ 1 ) ) = ( 1 / 2 ) ( 2 ⋅ 2 ) = ( 1 / 2 ) ( 4 ) = 2

1For example, if the function f ( x ) = x + 5 , then f − 1 ( x ) = x − 5 .

2 ∘ 2 = ( 1 / 2 ) ( ( 2 ⋅ 2 ) ⋅ ( 2 ⋅ 2 ) ) = ( 1 / 2 ) ( 4 ⋅ 4 ) = ( 1 / 2 ) ( 16 ) = 8 1

Example 2. Let us take the function f ( x ) = x + 1 và its inverse function f − 1 ( x ) = x − 1 . In this case, f T ( a ) = f ( a ) for all whole numbers a & f f T ( c ) = f − 1 ( c ) . This allows us lớn build the whole-number arithmetic A = 〈 W , ⊕ , ∘ 〉 and perform summation and multiplication in it finding some sums and products:

1 ⊕ 1 = ( ( 1 + 1 ) + ( 1 + 1 ) ) − 1 = ( 2 + 2 ) − 1 = 4 − 1 = 3

2 ⊕ 2 = ( ( 2 + 1 ) + ( 2 + 1 ) ) − 1 = ( 3 + 3 ) − 1 = 6 − 1 = 5

and

1 ∘ 1 = ( ( 1 + 1 ) ⋅ ( 1 + 1 ) ) − 1 = ( 2 ⋅ 2 ) − 1 = 4 − 1 = 3

2 ∘ 2 = ( ( 2 + 1 ) ⋅ ( 2 + 1 ) ) − 1 = ( 3 ⋅ 3 ) − 1 = 9 − 1 = 8

Example 3. Let us take the function f ( x ) = log 2 x and its inverse function f − 1 ( x ) = 2 x . This allows us lớn build the whole-number arithmetic A = 〈 W , ⊕ , ∘ 〉 and perform summation và multiplication, in it finding some sums & products:

1 ⊕ 1 = 2 ( log 2 1 + log 2 1 ) = 2 ( 0 + 0 ) = 2 0 = 1

2 ⊕ 2 = 2 ( log 2 2 + log 2 2 ) = 2 ( 1 + 1 ) = 2 2 = 4

while

1 ∘ 1 = 2 ( log 2 1 ⋅ log 2 1 ) = 2 ( 0 ⋅ 0 ) = 2 0 = 1

2 ∘ 2 = 2 ( log 2 2 ⋅ log 2 2 ) = 2 ( 1 ⋅ 1 ) = 2 1 = 2

Example 4. Let us take the function f ( x ) = x − 1 & its inverse function f − 1 ( x ) = x + 1 . In this case, f T ( a ) = f ( a ) for all whole numbers a and f T ( c ) = f − 1 ( c ) . This allows us lớn build the whole-number arithmetic A = 〈 W , ⊕ , ∘ 〉 & perform summation and multiplication in it finding some sums và products:

1 ⊕ 1 = ( ( 1 − 1 ) + ( 1 − 1 ) ) + 1 = ( 0 + 0 ) + 1 = 0 + 1 = 1

2 ⊕ 2 = ( ( 2 − 1 ) + ( 2 − 1 ) ) + 1 = ( 1 + 1 ) + 1 = 2 + 1 = 3

2 ⊕ 1 = ( ( 2 − 1 ) + ( 1 − 1 ) ) + 1 = ( 1 + 0 ) + 1 = 1 + 1 = 2

and

1 ∘ 1 = ( ( 1 − 1 ) ⋅ ( 1 − 1 ) ) + 1 = ( 0 ⋅ 0 ) + 1 = 0 + 1 = 1

2 ∘ 2 = ( ( 2 − 1 ) ⋅ ( 2 − 1 ) ) + 1 = ( 1 ⋅ 1 ) + 1 = 1 + 1 = 2

One more unusual property of Non-Diophantine arithmetics is related to lớn physics. Physicists often use relations a ≪ b , which means that a is much smaller than b, & b ≫ a , which means that b is much smaller than a. However, these relations vì not have an exact mathematical meaning và are used informally. In contrast, Non-Diophantine arithmetics provide rigorous interpretation & formalization for these relations. Namely Burgin, 1997 <28> ,

2The operator Å follows the standard rules of arithmetic & has no restrictions.

a ≪ b if & only if b ⊕ a = b 2

Note that this is impossible in conventional mathematics because for any number a > 0, the sum b + b is larger than b. At the same time, there are Non-Diophantine arithmetics, in which b ⊕ a = b is true for different a, b > 0. One of this type of arithmetic is considered in Example 4.

5. Conclusions

Expressions such as 1 + 1 = 3, 2 + 2 = 5, 2 + 2 = 3, 1 + 1 = 1 và 1 + 1 = 0 are symbolically used in economics and other realms of human activity. In addition, books exist which use these expressions as metaphors (Archibald, 2014 <29> ; Trott, 2015 <30> ; The Business Book, năm trước <31> ). Although for a long time these expressions were considered mathematically meaningless, we show that they are mathematically correct in some Non-Diophantine arithmetics or in prearithmetics. Therefore these expressions are approved and authorized by mathematical laws và are able to reflect phenomena in economics and physics.

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At the same time, rules of Non-Diophantine arithmetics are implicit in many realms of everyday life. Therefore, Non-Diophantine arithmetics are also becoming an integral part of sciences such as psychology, sociology and education, where they can explain và extend the known laws and principles and discover new ones.