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This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades. [55] He proved the associative and commutative properties, among others, through mathematical induction .

Defining (−2) + 1 using only addition of positive numbers: (2 − 4) + (3 − 2) = 5 − 6.
Further information: Integer

The simplest conception of an integer is that it consists of an absolute value (which is a natural number) and a (generally either positive or negative ). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases:

Although this definition can be useful for concrete problems, it is far too complicated to produce elegant general proofs; there are too many cases to consider.

A much more convenient conception of the integers is the Grothendieck group construction. The essential observation is that every integer can be expressed (not uniquely) as the difference of two natural numbers, so we may as well define an integer as the difference of two natural numbers. Addition is then defined to be compatible with subtraction:

Addition of rational numbers can be computed using the least common denominator , but a conceptually simpler definition involves only integer addition and multiplication:

As an example, the sum 3 4 + 1 8 = 3 × 8 + 4 × 1 4 × 8 = 24 + 4 32 = 28 32 = 7 8 {\displaystyle {\frac {3}{4}}+{\frac {1}{8}}={\frac {3\times 8+4\times 1}{4\times 8}}={\frac {24+4}{32}}={\frac {28}{32}}={\frac {7}{8}}} .

Computer engineering is a domain of engineering that has many subdomains. One of that subdomain is a stream that has vast future applications – AI or Artificial Intelligence, that need no introduction today. The chances are that your smartphone might have an AI assistant already helping you with your day to day tasks. Almost every industry needs a computer engineer be it on the hardware or software side.

Pursuing computer engineering also helps you to prove your worth in other prominent fields like robotics, programming, banking software development, etc. If you are into computers and programming; computer engineering holds many treasures for you to explore.

Computer programmers in US earn as high as $99K per year , and a majority of them claim a high level of job satisfaction.

If you find chemistry interesting and love to experiment with new ideas, this might be the one for you. Chemical engineering is now being a target by many aspiring students as the possibilities and scope are widening with every passing day. Chemical engineers contribute across differebt industry verticals including medical, oil and gas sector, battery development and environmental science.

Chemical engineers make up to $119K annually with an Tote Bag Interior Imagination 6 by VIDA VIDA mkYrs9FNee

We have talked about how the software is going to revolutionize the world, but every software needs a platform to work. This is where the harmony of hardware and software happens. With the proliferation of Internet of Things (IoT) and smart devices, electronics will continue to play a dominant role in our lives. The possibilities are actually endless! Mobiles, computers, servers, structural electrical, and the list goes on and on.

The average salary of an Electronics Engineer in US is $77K .

Biomedical engineering field grants more than just a hefty paycheck to the engineers, they provide increased job satisfaction. This is because biomedical engineers are researching and working on various medicines that would make the human race one step closer to leading a healthy life. This is why 75% of the biomedical engineers have reported that their job makes them feel better about themselves.

Having a higher meaning to the job that you are doing makes it easy to love what you are doing. If you want to be an engineering graduate who wants to help the world become a better place specifically through the work you are doing, Biomedical Engineering is the trade you should opt for!

Biomedical engineers have the opportunity to become medical scientists. Then again paths like neuromodulation services, biomaterials, and stem cell engineering are a few of the highest paying jobs out there.

17 years after the FCPA enacting, a Parliamentary Financial Commission in Bonn presented a comparative study on legal corruption in industrialized OECD countries [105] As a result, they reported that in most industrial countries even at that time (1994) foreign corruption was legal, and that their foreign corrupt practices had been diverging to a large extent, ranging from simple legalization, through governmental subsidization (tax deduction), up to extremes like in Germany where foreign corruption was fostered, whereas domestic was legally prosecuted. Consequently, in order to support national export corporations the Parliamentary Financial Commission recommended to reject a related previous Parliamentary Proposal by the opposition leader which had been aiming to limit German foreign corruption on the basis of the US FCPA. [106] Only after the OECD Anti-Bribery Convention came into force, did Germany withdraw the legalization of foreign corruption in 1999. [107]

The Foreign corrupt practices of industrialized OECD countries 1994 (Parliamentary Financial Commission study, Bonn). [105]

Belgium: bribe payments are generally tax deductible as business expenses if the name and address of the beneficiary is disclosed. Under the following conditions kickbacks in connection with exports abroad are permitted for deduction even without proof of the receiver:

In the absence of the required conditions, for corporate taxable companies paying bribes without proof of the receiver, a special tax of 200% is charged. This special tax may, however, be abated along with the bribe amount as an operating expense.

Denmark: bribe payments are deductible when a clear operational context exists and its adequacy is maintained.

France: basically all operating expenses can be deducted. However, staff costs must correspond to an actual work done and must not be excessive compared to the operational significance. This also applies to payments to foreign parties. Here, the receiver shall specify the name and address, unless the total amount in payments per beneficiary does not exceed 500 FF. If the receiver is not disclosed the payments are considered "rémunérations occult" and are associated with the following disadvantages:

Japan: in Japan, bribes are deductible as business expenses that are justified by the operation (of the company) if the name and address of the recipient is specified. This also applies to payments to foreigners. If the indication of the name is refused, the expenses claimed are not recognized as operating expenses.

Canada: there is no general rule on the deductibility or non-deductibility of kickbacks and bribes. Hence the rule is that necessary expenses for obtaining the income (contract) are deductible. Payments to members of the public service and domestic administration of justice, to officers and employees and those charged with the collection of fees, entrance fees etc. for the purpose to entice the recipient to the violation of his official duties, can not be abated as business expenses as well as illegal payments according to the Criminal Code.

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