Multiplication is repeated addition.

2x3= 2+2+2 AxB= A+A B times.

Raising to a power is repeated multiplication.

2^3= 2x2x2 A^B = AxA B times

Roots are the inverse of powers

Division is the inverse of multiplications.

Given this… the OC’s comment about giving exceptions to powers in the rules is nonsense because they are still operations.

▲

mojomark a day ago | parent [-]

Multiplication is not the same as repeated addition, which you can observe as soon as you try the two different operations on fractions or negative numbers. For example, you can't add a number to itself 1/3 times, but you can certainly scale a number by 1/3.

Multiplication is a scaling operation.

▲

reichstein a day ago | parent | next [-]

Multiplication is the generalization of repeated addition. Repeated addition can be generalized to fractional values.

In the integers, it is the same thing, with a rule for negative times.

In the rational numbers, you can generalize to adding a number to itself a rational number of times. Adding a/b to itself p/q times is defined as adding a to itself p times, with a divisor that is adding b to itself q times.

So adding the rational 5 (5/1) with itself 2_and-1/3 (4/3) times just means (5+5+5+5) / (1 + 1 + 1) = 20/3, a rational that we can also write as 6 -and-2/3. Only addition of integers is used.

Which is the same as rational multiplication (a/b)(p/q) = (ap)/(b*q).

But it's true that you can't say that rational multiplication is the same as _integer_ repeated addition.

▲

goodmythical a day ago | parent | prev | next [-]

Can you not? 9+(9(1/3)) [adding nine to itself a third of a time] is nine plus one third of nine {or nine divided by three} which is one of the three in three times three that equals nine which is three therefore 9+(9(1/3))=11 because 9+3=11

	▲	mojomark 19 hours ago \| parent \| next [-]
		By dividing by 3 you are introducing a scaler into the operation. Multiplication also flips signs (direction). That's why a negative * negative = positive, whereas a negative + negative = negative. The addition operator does not flip direction. Multiplication is not generally the same as multiple addition operations. However, it has the same result for positive whole numbers. I am not writing this to flippant. I'm a 40-something year old engineer and I litterally thought the same thing until a year ago-ish. I was trying to build a function (I can't recall what exactly) for a project and I said, well instead of multiplying these terms (which wasn't efficient for what I was doing or something), I'll just cycle through and add the terms X-number of times. ChatGPT said "uh, no, that's not the same thing," to which I said NFW and proceeded down a deep rabbit hole on the difference. I'm glad I did because I also now have a much better intuition for imaginary numbers, which I was severely lacking before.
	▲	red369 21 hours ago \| parent \| prev \| next [-]
		:) I like it when people who are clearly smart do things like this! Makes me feel better. Reminds me of this story: https://news.ycombinator.com/item?id=5663709
	▲	srean 20 hours ago \| parent \| prev \| next [-]
		That's Grothendieck multiplication.
	▲	alpaca128 a day ago \| parent \| prev [-]
		9+3=12, but otherwise I agree

▲

anthk 16 hours ago | parent | prev [-]

>you can't add a number to itself 1/3 times,

Learn logarithms.