# Thread: do the rules of arithmetic apply to real numbers?

1. multiplication is defined in arithmetic "repetition of addition". If we multiply 2 real numbers say: 1.4242 .... * 2.7818.... can we obtain same result by repetition of addition?
thanks

2.

3. If one defines multiplication in that way, then one has indeed ruined multiplication in the real numbers.

4. Originally Posted by PhysBang
If one defines multiplication in that way, then one has indeed ruined multiplication in the real numbers.
that is not a reply, Phys, if you know, you tell, please.
that is the current definition. what is then definition for real number?, the question was: how do you multiply 2 real numbers? usually you elude the problem writing e*sqrt2, but if you want to perform the operation what do you do? I knew that you just round them to the required approximation, dont'you?

5. If you want a numerical answer for any arithmetic problem involving irrational numbers, you will (with some exceptions) have to round or truncate. This applies to addition, etc. as well as multiplication.

All it does is make you look like a ranting fool.

7. Originally Posted by mathman
If you want a 2)numerical answer for any arithmetic problem involving 3)irrational numbers, you will (with some 1)exceptions) have to round or truncate. This applies to addition, etc. as well as multiplication.

1) give an example of those exceptions, and please tell me:
2) what can be there besides a numerical answer
3) why there is a distinction between an irrational 1.4142... and a rational periodic number such as 1/3 = 0.33333..., they are both infinite decimal numbers, don't we have same problem there?
thank you very much.

(P.S.you are improving your manners, meteor, thanks for not using foul language, could you begin to say something sensible,now?)

8. Originally Posted by ray
Originally Posted by mathman
If you want a 2)numerical answer for any arithmetic problem involving 3)irrational numbers, you will (with some 1)exceptions) have to round or truncate. This applies to addition, etc. as well as multiplication.

1) give an example of those exceptions, and please tell me:
2) what can be there besides a numerical answer
3) why there is a distinction between an irrational 1.4142... and a rational periodic number such as 1/3 = 0.33333..., they are both infinite decimal numbers, don't we have same problem there?
thank you very much.

(P.S.you are improving your manners, meteor, thanks for not using foul language, could you begin to say something sensible,now?)
1) e - e = 0, e^(πi) = -1
2) symbolic answer - the diagonal of a unit square has length √2.
3) .333.... can be expressed with a finite number of numerical symbols (numerals and /) 1/3. Irrational numbers cannot.

9. Originally Posted by mathman

1)a) e - e = 0,
e^(πi) = -1
2) symbolic answer - the diagonal of a unit square has length √2.
3) .333.... can be expressed with a finite number of numerical symbols (numerals and /) 1/3. Irrational numbers cannot.

[if you allow me,
1) you have chosen particular examples, that produce finite numbers,
a) any irrational minus itself is 0
2) I see no difference betwen a symbol and its decimal expansion :sqrt 2 and 1.4142... you use dots to say it is irrational, then it becomes a [more explicit/analytical symbol]
3) there is no difference, too, between sqrt 2 and 1/3 : formally they are both symbols, math-wise they are simply non-performed operations:
the square root of 2 is = ?, no result; one divided by three is = ? no result. So linguistically, formally, math-ly there seems to be no difference, substantially they are both decimal numbers with an infinite number of decimals. Am I wrong? this seems the only conceptual-math difference:
0.333...rationals are result of impossible division and
1.4142... irrationals are results of impossible sqrt of non-perfect-square-numbers. Do you agree?] if you choose to not reply to this:

Now back to OP, you said that we want to perform an operation with reals we must round them down, that is: truncate them, that is perfectly right, so if we multiply e = 2.7818 *sqrt 2 = 1.4142 we get 3.93402156 .
4) Can we say now this multiplication is repetition of addition 1.4142+1.4142+ 1.4142*0.7818? or what?

5) if you are also an expert on set theory: are reals where included in original ZF axiom theory? it is included in Cantor's "on a characteristic property of all real algebraic numbers"? do you know why real numbers ar called algebraic and not arithmetical?

Thanks a lot again, mathman, you are great help!

10. Isn't 2.7818.... being added to itself 1.4242 .... times?

11. To ray: What is your point?

Real numbers are a much larger set than algebraic numbers. Algebraic numbers (roots of polynomial equations with integer coefficients) are countable, real numbers are not.

12. Originally Posted by ray

[if you allow me,
1) you have chosen particular examples, that produce finite numbers,
a) any irrational minus itself is 0
2) I see no difference betwen a symbol and its decimal expansion :sqrt 2 and 1.4142... you use dots to say it is irrational, then it becomes a [more explicit/analytical symbol]
3) there is no difference, too, between sqrt 2 and 1/3 : formally they are both symbols, math-wise they are simply non-performed operations:
the square root of 2 is = ?, no result; one divided by three is = ? no result. So linguistically, formally, math-ly there seems to be no difference, substantially they are both decimal numbers with an infinite number of decimals. Am I wrong? this seems the only conceptual-math difference:
0.333...rationals are result of impossible division and
1.4142... irrationals are results of impossible sqrt of non-perfect-square-numbers. Do you agree?] if you choose to not reply to this:

Now back to OP, you said that we want to perform an operation with reals we must round them down, that is: truncate them, that is perfectly right, so if we multiply e = 2.7818 *sqrt 2 = 1.4142 we get 3.93402156 .
4) Can we say now this multiplication is repetition of addition 1.4142+1.4142+ 1.4142*0.7818? or what?

5) if you are also an expert on set theory: are reals where included in original ZF axiom theory?
6) if not, when where they added, and complex numbers?

no point, mathman, just 6 questions. answer the ones you wisk or give me link. Thanks

13. 1) e*π is irrational.
2) so what?
3) these is a difference. Because you dislike 1/3, it doesn't make it any less meaningful.
4) multiplication is initially defined as repetition of addition. However its gets generalized when we allow fractions and further generalized for reals and complex numbers.
5) I have never formally gone from ZF to get numbers. Numbers are defined by starting with integers, then rationals, then real.
6) Complex numbers are included by giving meaning to √(-1).

14. The definition given is not the proper definition of multiplication. The multiplication of any number by any number greater than two can be defined as the repetition of addition. Any real number multiplied by any real number above two including irrational numbers will fit the aforementioned definition ie "1 x 3.3333....," is one + one repeated one and one-third times after the initial instance. "1 x 2," is only one instance of the addition of one and one and therefore cannot be described as having been repeated other than having been repeated zero times, which of course is not to have been repeated at all.

A true definition of multiplication is a mathematical operation, symbolized by a × b, ab, ab, or ab, and signifying, when a and b arepositive integers, that a is to be added to itself as many times as there are units in b; the addition of a number toitself as often as is indicated by another number, as in 2×3or 5×10.

This problem simply required an English major not a math major.

15. Originally Posted by FutureBeast
The definition given is not the proper definition of multiplication. The multiplication of any number by any number greater than two can be defined as the repetition of multiplication..
your post requires no comment, it is self-destroying

16. Originally Posted by ray
your post requires no comment, it is self-destroying
Insult if you must. Thank you for pointing out my mistype, it allowed me to fix it. I will not ask you to defend your mischaracterization of multiplication since it is evident that you don't care.

17. Originally Posted by FutureBeast
Originally Posted by ray
your post requires no comment, it is self-destroying
Insult if you must. Thank you for pointing out my mistype, it allowed me to fix it. I will not ask you to defend your mischaracterization of multiplication since it is evident that you don't care.
I was not insulting, sorry if you misunderstood, it is not a mistype. btw you were snarky, why are you whining?.
I do not understand why you are quibbling, flogging a dead horse. Definition of multiplication is repetition of addition. full stop. period. google it, you have a million links. what is your point?

18. Originally Posted by ray
I was not insulting, sorry if you misunderstood, it is not a mistype. btw you were snarky, why are you whining?.
I do not understand why you are quibbling, flogging a dead horse. Definition of multiplication is repetition of addition. full stop. period. google it, you have a million links. what is your point?
I just disagree with the definition given. It is inaccurate. Is 1(.03) the repetition of addition?

19. Multiplication by integers can be defined as repeated addition. To go beyond requires various extensions, first to rationals, then to reals, and complex.

From Wikipedia:

Multiplication (often denoted by the cross symbol "×") is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic (the others being addition, subtraction and division).
Because the result of scaling by whole numbers can be thought of as consisting of some number of copies of the original, whole-number products greater than 1 can be computed by repeated addition; for example, 3 multiplied by 4 (often said as "3 times 4") can be calculated by adding 4 copies of 3 together:
Here 3 and 4 are the "factors" and 12 is the "product".
There are differences amongst educationalists as to which number should normally be considered as the number of copies and whether multiplication should even be introduced as repeated addition.[1] For example 3 multiplied by 4 can also be calculated by adding 3 copies of 4 together:
Multiplication of rational numbers (fractions) and real numbers is defined by systematic generalization of this basic idea.

20. Originally Posted by ray
the question was: how do you multiply 2 real numbers? usually you elude the problem writing e*sqrt2, but if you want to perform the operation what do you do? I knew that you just round them to the required approximation, dont'you?
back to square one, mathman, that is my original question, how do you generalize for real [infinite (rational/irrational)] or complex, numbers?
For reals it is difficult, for complexes seems impossible. Your link"generalization" says for reals is the limit, and for complexes ..when imaginary part is zero.This is cheating.

21. Originally Posted by ray
Originally Posted by ray
the question was: how do you multiply 2 real numbers? usually you elude the problem writing e*sqrt2, but if you want to perform the operation what do you do? I knew that you just round them to the required approximation, dont'you?
back to square one, mathman, that is my original question, how do you generalize for real [infinite (rational/irrational)] or complex, numbers?
For reals it is difficult, for complexes seems impossible. Your link"generalization" says for reals is the limit, and for complexes ..when imaginary part is zero.This is cheating.
To get from rationals to reals, set up sequences of rationals which converge to each term of the reals, then the limit of the product is the product of the limit.

For complex, it is trivial (a+bi)(c+di) = ac - bd + (ad + bc)i., where a,b,c,d are real.

22. where is repetition of addition?

23. Buried under many layers of additional structure, after all I doubt anyone wants to add 3.621 to itself 2.893 times. Bare with me and i'll see if i can dig through this additional structure for you.

As mathman stated the real numbers are just the limit of a series of rational numbers, if the series is finite then we have another rational number otherwise the number is irrational, the product of two real numbers is then the limit of a series of products of rational numbers. So multiplication of reals has now been reduced to the addition of many products of rational numbers. The product of two rational numbers is a rational number who's numerator and denominator are the products of two integers that constitute the numerator and the denominator of the numbers we are taking the product of. (that's a mouthful but i'm sure you know how two rational numbers are conventionally multiplied together) So the product of two rational numbers is another rational number with a numerator and denominator that is determined by the multiplication of integers. Integer multiplication being defined in terms of repetition of addition.

24. Originally Posted by mathman
*If you want a numerical answer for any arithmetic problem involving irrational numbers, you will (with some exceptions) have to round or truncate. This applies to addition, etc. as well as multiplication.
Originally Posted by wallaby
As mathman stated *
**The product of two rational numbers is a rational number who's numerator and denominator are the products of two integers that constitute the numerator and the denominator of the numbers we are taking the product of. (that's a mouthful but i'm sure you know how two rational numbers are conventionally multiplied together) So the product of two rational numbers is another rational number with a numerator and denominator that is determined by the multiplication of integers. Integer multiplication being defined in terms of repetition of addition.
wallaby, mathman stated that*. I offered to take that to its natural conclusion. Whay cannot we be consequent? why must we deceive ourselves?what is a limit if not a round up/down ? [btw truncation is not rounding?what is the difference?] isn't that cheating?
you show me what does that** concretely mean different from: 2.718281828458......x 1.414213562..... = 3.844231028 / 3.844231028158 .full stop, period

25. Ray:
The only thing I can make out of this is that you are concerned about writing down a number (or a computer representation) where the number itself cannot be expressed that way because it is irrational or a repeating decimal. That is a legitimate computer science concern, but not a mathematics concern.

26. Originally Posted by ray
wallaby, mathman stated that*. I offered to take that to its natural conclusion. Whay cannot we be consequent? why must we deceive ourselves?what is a limit if not a round up/down ? [btw truncation is not rounding?what is the difference?] isn't that cheating?
you show me what does that** concretely mean different from: 2.718281828458......x 1.414213562..... = 3.844231028 / 3.844231028158 .full stop, period
mathman stated
Originally Posted by mathman
To get from rationals to reals, set up sequences of rationals which converge to each term of the reals, then the limit of the product is the product of the limit.

For complex, it is trivial (a+bi)(c+di) = ac - bd + (ad + bc)i., where a,b,c,d are real.
Originally Posted by ray
To which i responded.

If your concern is with numerical calculation only, i.e. the question of how we get computers to do mathematics, then since computers only have a finite precision with which to represent numbers (obviously) the representation is a truncation of the actual value. Computers cannot round to the last value since they cannot compute the last value, either way the value is in error of the real value so it's not cheating one way or another.

In regards to your somewhat demanding request that i elaborate further on my explanation, using the example of . The rational expansions of these numbers is given below.

In this case the index 'n' is an integer, thus each term in the series is a rational number. Multiplying these two series together gives,

So were we capable of performing these summations an infinite number of times we could get an exact answer. If we instead use a computer we would truncate the series to a finite number of summations and obtain an approximate answer.

27. Originally Posted by wallaby
So were we capable of performing these summations an infinite number of times we could get an exact answer..
and you are not

28. Originally Posted by ray
and you are not
Irrelevant, despite the fact that we are not capable of summations of infinite numbers of terms does not render the point invalid. The numbers which we are multiplying together, 'e' and are irrational and can only be expressed exactly as such series. Otherwise we are not solving the problem of but rather a similar problem based on multiplying approximations of those numbers.

29. not expressed, imagined. so what?

30. Does anyone disagree that this thread has run its course?

If not, I propose locking it. Any objections other than those of ray?

31. Originally Posted by Guitarist
Does anyone disagree that this thread has run its course?

If not, I propose locking it. Any objections other than those of ray?
I agree. ray seems to beating a dead horse, but he is not satisfied that it is dead.

32. Since the only responses ray deems appropriate are dismissive and without qualification, i'm done trying.

33. Originally Posted by Guitarist
Does anyone disagree that this thread has run its course?

If not, I propose locking it. Any objections other than *those of ray?
that* is a false assumption, guitarist, I have no objections to you closing the thread, as the leading mathematicians have run out of arguments, and contradict themselves.
I have an objection to your statement that the thread has run out of course, because I remained coherent with OP:how is multiplication of reals repetition of addition?
I have a further objection , too. An objection in principle which I pass on to Administration .Closing a thread, forcibly closing discussion in a forum is a contradictio in terminis, an oxymoron, an absurdity. In this case it is mercy-killing because discussion is agonizing

34. I do hope that the role of limits in the definition and properties of real numbers is generally (if not, alas, universally) recognised.

35. Originally Posted by agingjb
I do hope that the role of limits in the definition and properties of real numbers is generally (if not, alas, universally) recognised.
I do recognize it, the point is that a) you are not capable of making summations an infinite number od times, b) when or if you jump to the limit you are surreptisiouly rounding.
I am sure you are familiar with the proof that x=0.9999999999...= 1,[10x-x= 9x =9] that is exactly what you are doing here.
9x makes 8.999999......1, you may round it openly : 8.99999...1= 9 or you may play with infinity. Do you read me?

36. The axiomatic development of real numbers depends on the limit concept precisely in order to avoid completed infinities. This is not the place to repeat the extensive literature of real analysis (Landau, Spivak, many others). People are, of course, free to read and reject that literature, although my advice would be to accept its insights as being essential to the structure of a considerable body of mathematics.

37. Originally Posted by ray
I remained coherent with OP:how is multiplication of reals repetition of addition?
And "coherently" wrong - though here we would say "consistently". You have been proved wrong in your OP too often for any further discussion to be useful

An objection in principle which I pass on to Administration .
Yes, please do that - all moderators are answerable to the Admins after all. If I am out of line, I shall be told, have no fears about that
Closing a thread, forcibly closing discussion in a forum is a contradictio in terminis, an oxymoron, an absurdity.
I suggest you look up the true definition of oxymoron - the usual definition is not applicable here.