i am analysis a book in discrete benidormclubdeportivo.orgematics and also it assumes the a multiplication of two integers yields an integer.

Although the this book"s speak is justifiable due to the fact that the publication is do an assumption, I found that the this is "completely wrong however it is still a great estimation".

let \$n ge 1\$ it is in an integer. Then, by the over assumption, \$(n^2-3) * 5\$ yields an integer. But, this is not accurate.

Factorize \$n^2-3 = (n+sqrt3) (n-sqrt3)\$.

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Now, usage a computer system system to evaluate this statement substituting any integer (\$1\$ or larger). Friend will uncover that, extremely, the result is not an integer and therefore us kinda have an contradiction between \$n^2-3\$ and also its factorization.

So, "Is it accurate to say the multiplication of 2 integers returns an creature ?"  You room confusing computer arithmetic with benidormclubdeportivo.orgematical arithmetic. In a computer, creature arithmetic is precise as long as girlfriend don"t overflow and also \$5(n^2-3)\$ will always yield an exact integer if \$n\$ is an integer and the computation is done utilizing integer representations. When you go the end of the integers, you use floats, which have actually a limited number of bits the precision. It might be the \$(sqrt 3)^2 eq 3\$ in a computer, yet it is constantly true in benidormclubdeportivo.org the \$(sqrt 3)^2 = 3\$. In benidormclubdeportivo.orgematics it is accurate to say the multiplying (or subtracting or adding) two integers yields an integer. In a computer system using 32 bit integers, \$2^20 cdot 2^20\$ will give an overflow, no an integer. Computer system arithmetic is close to real arithmetic, but it has some foibles to catch the unwary. If your computer system system does not offer an integer when computer \$(n-sqrt3)(n+sqrt 3)\$, climate it is because of the floating allude arithmetic associated in computer with the irrational number \$sqrt3\$. Watch the ar in https://en.m.wikipedia.org/wiki/Floating_point on "accuracy problems."

This is a an excellent lesson in realizing the the results of computer devices space not past question, and that they have limitations.

Of course the product of 2 integers is one integer. Via the distributive law, you have the right to view multiplication as recurring addition, and also hopefully girlfriend believe including integers gets you integers. You need to step back and asking yourself, "what is one integer"? girlfriend can"t decide if other is an integer or not till you define the integers!

Defining the integers requires picking axioms the integers must satisfied. Choose the "right" axioms so the the an interpretation is consistent, and unambiguous (so that, for instance, you can tell the difference in between the integers and modular arithmetic, or polynomials through integer coefficients) is a little tricky, but has been worked out in detail and several choices exist.

Some axiomatizations that the integers define the to have actually ring structure, which implies they space closed under addition and multiplication. If you choose these axioms, the truth that two integers main point to an essence is automatic.

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Other axiomatizations (for circumstances those occurring from prolonging the Peano axioms) perform not include an axiom about closure the the integers under multiplication, yet instead offer a recursive meaning for the product of 2 integers, and enough additional axioms (i.e. Induction) come prove that the product of two integers is constantly an integer.