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Gigantic_Idiot t1_iybpgt8 wrote

Reply to comment by BurnOutBrighter6 in ELI5 why we first multiply, then add by TheManNamedPeterPan

Adding on since I don't have enough for a parent comment, but multiplication is doing an addition multiple times. So the example could also be written as 1+2+2+2.

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Way2Foxy t1_iybsn7w wrote

>multiplication is doing an addition multiple times.

Only for integer multiplication, though.

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Sereaph t1_iybyrjj wrote

No, the rules apply through the real numbers. It's just easier to visualize in integers.

Let's take 1.4 × 2.5. Both are non-integers.

You can still interpret this as taking 1.4 and adding it to itself 2.5 times: 1.4 + 1.4 + 0.7 = 3.5

Another example, take pi, an irrational number. pi × pi = pi + pi + pi + (.1415...)pi = 9.8696...

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Way2Foxy t1_iybyvh8 wrote

Sure, but you're kicking the can, no? Define that last term in pi^(2),

(.1415...)pi

using only your repeated addition definition.

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Sereaph t1_iybz0eq wrote

It's simply a fraction of pi. You don't need to think of the parenthesis as multiplication per se, just that it's a modifier that represents the leftover.

You can say "half of 4 is 2", "a quarter of 12 is 3". In a similar vein, .1415...th of pi is 0.44...

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Way2Foxy t1_iybz77k wrote

Fractions are inherently using multiplication, though. I'm not saying you can't visualize it as repeated addition, just that it's not equivalent to repeated addition, outside of integers (or at least having one of the numbers being an integer).

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Sereaph t1_iybzgmy wrote

No, but it IS still equivalent to repeated addition. It just gets more complicated in how that's represented symbolically. The fundamental operation doesn't change just because we use less clean symbology of rational and irrational numbers. The concept is still the same. Multiplication is an operation upon addition.

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Way2Foxy t1_iyc036j wrote

If it is equivalent, then please give me the exact value of 0.75 x 0.44, only using addition.

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Phrygiaddicted t1_iyc18ek wrote

just do long multiplication. this reduces the problem to repeated integer multiplications.

its also equivlent to his example of adding up the integer parts then you deal with the fractional part by multiply everything so that the fractional part is integral, doing the repeated addition then dividing the answer at the end by the factor you multiplied by. this is equvalent to shifting the decimal point. the point doesn't change how you approach the algorithm, and the answer in the end is still a rational fraction, not really a "single number" 0.25 isnt simpler than 1/4, its just another way to write it.

that trick wont work for irrationals though. but they aren't calculable no matter what you do so...

this is really trivial for binary multiplication, as it just reduces to shifting the number up one place across itself and adding the digits to itself IF the digit in that place is a 1; as you either add 0xthat (which is no addition) or 1xthat, (which is no multiplication). binary addition is also simple, it reduces to looking if they digits are different, in which case its 1, otherwise its 0 and u carry a 1 if theyre both 1.

    1011 (11)
   x0110 (6)
   -----
   1011  (22)
  1011   (44)
  ----
 1000010 (66)

>give me the exact value of 0.75 x 0.44, only using addition.

as for your question... we can agree integer multiplcation is trivial and can always reduce to repeted addition so i wont write it out but...

4x5 + 4x70 + 40x5 + 40x70 = 3300; put decimal place in correct 2x2=4 point position = 0.33

all of this works with rational numbers and they reduce to repeated addition. his example with pi is bogus for different reasons, and that's just because irrational and especially transcendental numbers like pi are just uncalculable; so you'd end up doing this long multiplication process forever; infact you wouldnt even get to the point of multiplying pi, because you're still working out what pi is so you can multiply it. all calculations that have ever been done are on rational numbers. "real" numbers are just symbols we manipulate and replace with approximations when a calculation is needed. they aren't real.

or you can just pretend its 22/7.

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Sereaph t1_iyc1wc1 wrote

you're trying to be clever using those decimals, but the concept is still the same.

First let's convert those decimals into symbology that's easier to work with (fractions):

0.75 = 3/4

0.44 = 11/25

0.75 x 0.44 = 3/4 x 11/25

So therefore using ADDITION to evaluate the multiplication of the fractions,

11+11+11 = 33 for the numerator

25+25+25+25 = 100 for the denominator

The fraction 33/100 simplifies to 0.33 in decimal form.

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Way2Foxy t1_iyc2s6k wrote

I wasn't trying to be clever, I was just choosing random numbers.

You're right, though, and you definitely can use long multiplication to multiply any two rationals using only addition (though not sure if/how you can reduce to the lowest terms).

I got lost/distracted and started arguing something I didn't mean to initially. My initial argument that I still hold is that you can't define multiplication as simply repeated addition, and to further clarify I mean strictly that if you multiply a and b you add a to itself b-1 times.

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thatnotsorichrichkid t1_iybxyrj wrote

Can you elaborate? I feel like the logic should apply to all numbers. Hell even in the complex sphere you should be able to do multiplication through iterations of addition.

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Way2Foxy t1_iyby7kg wrote

Well, how would you describe pi*e in terms of iterative addition?

If I were to try, I'd say e+e+e+(pi-3)e, but that's just shifting the multiplication somewhere else, no?

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thatnotsorichrichkid t1_iyd6k6r wrote

That's exactly how I'd do it, but i kinda understand your point. To do a set of non-whole integers multiplied, one requires a way of splitting 'the last plus', and the only way my Imagination allows me to would be multiplication

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