I think it's okay to say: "It's useful to visualize (or think of) multiplication as repeated addition, in contexts where that makes sense." Properly qualified the statement seems true and unbojectionable, and I think it's a useful way to help students understand multiplication. (It's not the only way.)
The reservation I'd have with the statement "Multiplication is repeated addition" is that the use of the word "is" implies an identification, or a necessary derivation of one from the other. The operations aren't identical, and one is not necessarily derived from the other. As operations, they're distinct.
(The issue also has nothing per se to do with units. When you introduce units of length or area, you're using numbers in a particular applied context. Outside of that context, numbers don't have units.)
In everyday usage "addition" and "multiplication" are conventional designations. Conventions always have ambiguities and edge cases.
The most common formal structure in which you have both addition and multiplication is in a ring. A ring is a set with two operations, which we call "addition" and "multiplication" --- but those are just names. We could just as well have used "foo" and "bar". We tend to use the standard addition and multiplication symbols for those two operations, but that is also just a convention/convenience. (Note also that those operations are not unique.)
Rings include the integers, rationals, reals, complex numbers - but also (for instance) polynomials with real coefficients, 42 x 42 matrices with real entries, quaternions, or finite fields.
Now consider what happens as you list the axioms for a ring. You say that addition is associative, addition has an identity element (conventionally denoted "0"), every element has an additive inverse, and addition is commutative. That's addition.
Then you say multiplication is associative (and nowadays, since people find it convenient to assume this) and multiplication has an identity element (conventionally denoted "1").
At the moment, you have two independently defined operations, which have nothing to do with one another. "Multiplication" is therefore not identical to (or defined as) repeated addition. But it's not too useful to have two independent unrelated operations. You connect the two operations by introducting the distributive axiom: For all a, b, c in your ring,
a * (b + c) = a * b + a * c and (a + b) * c = a * c + b * c.
(scythe pointed out the importance of the distributive law in another reply.) The distributve axiom is huge! Consider
a * (b + c) = a * b + a * c.
From left to right, it says that "you can multiply out"; from right to left, it says that "you can take out a common factor", which are both standard operations in algebra.
Once the distributive property is available to connect the operations, it explains why you can "think of" multiplication as repeated addition: For instance,
3 * 2 = 3 * (1 + 1) = 3 * 1 + 3 * 1 = 3 + 3.
So you say "3 times 2" can be thought of as "3 added to itself 2 times". It does not say that multiplication is repeated addition, if "is" means "defined as" or "derived from",
Or consider one of the standard illustrations given to kids in grade school:
3 * 2 is $ $ $ which is [ $ $ $ ]
$ $ $ [ $ $ $ ]
This is 2 groups of 3, i.e. 3 + 3. So 3 * 2 = 3 + 3.
But what happened here? We relied on our physical intuition to "know" that putting the original 6 dollars into 2 bags in the second step didn't change the number of dollars. But formally, it is
3 * 2 = 3 * (1 + 1) = 3 * 1 + 3 * 1,
which is the distributive law again.
(BTW regrouping the dollars again into 3 groups of 2 is a standard way of motivating commutativity of multiplication, since 3 groups of 2 is visibly "the same as" 2 groups of 3.)
(Someone might suggest from the dollars example that multiplication might have arisen historically as a shorthand for repeated addition. I don't know math history well enough to say, but historical derivation doesn't imply identity or logical derivation.)
So I think saying "Multiplication is repeated addition" is a little sloppy in the use of the word "is", but we can agree to disagree about how much sloppiness is okay. The statement is fine as a way of giving students one (of many) ways to think about multiplication. (Pictures are important, too!)
Here's something to think about. (I don't have an opinion myself.) Suppose we have a complex multiplication: (7 i)(3 + 4 i). Following the interpretation above, I describe this as "7 i added to itself (3 + 4 i) times". If you don't like the sound of that ... why? You might say "You can't have '3 + 4 i' things." Well, in the real world discrete "things" come in nonnegative integer quantities ... or do they? For instance, we could make an agreement that "-3 things" means "I'm missing 3 things" or "you owe me 3 things". Nothing stops us from making agreements about the use of words. So maybe we could "agree" that "(3 + 4 i) times" means exactly an occurrence of the expression "3 + 4 i" in a context like this one. Is there any harm in that? :-)
The reservation I'd have with the statement "Multiplication is repeated addition" is that the use of the word "is" implies an identification, or a necessary derivation of one from the other. The operations aren't identical, and one is not necessarily derived from the other. As operations, they're distinct.
(The issue also has nothing per se to do with units. When you introduce units of length or area, you're using numbers in a particular applied context. Outside of that context, numbers don't have units.)
In everyday usage "addition" and "multiplication" are conventional designations. Conventions always have ambiguities and edge cases.
The most common formal structure in which you have both addition and multiplication is in a ring. A ring is a set with two operations, which we call "addition" and "multiplication" --- but those are just names. We could just as well have used "foo" and "bar". We tend to use the standard addition and multiplication symbols for those two operations, but that is also just a convention/convenience. (Note also that those operations are not unique.)
Rings include the integers, rationals, reals, complex numbers - but also (for instance) polynomials with real coefficients, 42 x 42 matrices with real entries, quaternions, or finite fields.
Now consider what happens as you list the axioms for a ring. You say that addition is associative, addition has an identity element (conventionally denoted "0"), every element has an additive inverse, and addition is commutative. That's addition.
Then you say multiplication is associative (and nowadays, since people find it convenient to assume this) and multiplication has an identity element (conventionally denoted "1").
At the moment, you have two independently defined operations, which have nothing to do with one another. "Multiplication" is therefore not identical to (or defined as) repeated addition. But it's not too useful to have two independent unrelated operations. You connect the two operations by introducting the distributive axiom: For all a, b, c in your ring,
(scythe pointed out the importance of the distributive law in another reply.) The distributve axiom is huge! Consider From left to right, it says that "you can multiply out"; from right to left, it says that "you can take out a common factor", which are both standard operations in algebra.Once the distributive property is available to connect the operations, it explains why you can "think of" multiplication as repeated addition: For instance,
So you say "3 times 2" can be thought of as "3 added to itself 2 times". It does not say that multiplication is repeated addition, if "is" means "defined as" or "derived from",Or consider one of the standard illustrations given to kids in grade school:
This is 2 groups of 3, i.e. 3 + 3. So 3 * 2 = 3 + 3.But what happened here? We relied on our physical intuition to "know" that putting the original 6 dollars into 2 bags in the second step didn't change the number of dollars. But formally, it is
which is the distributive law again.(BTW regrouping the dollars again into 3 groups of 2 is a standard way of motivating commutativity of multiplication, since 3 groups of 2 is visibly "the same as" 2 groups of 3.)
(Someone might suggest from the dollars example that multiplication might have arisen historically as a shorthand for repeated addition. I don't know math history well enough to say, but historical derivation doesn't imply identity or logical derivation.)
So I think saying "Multiplication is repeated addition" is a little sloppy in the use of the word "is", but we can agree to disagree about how much sloppiness is okay. The statement is fine as a way of giving students one (of many) ways to think about multiplication. (Pictures are important, too!)
Here's something to think about. (I don't have an opinion myself.) Suppose we have a complex multiplication: (7 i)(3 + 4 i). Following the interpretation above, I describe this as "7 i added to itself (3 + 4 i) times". If you don't like the sound of that ... why? You might say "You can't have '3 + 4 i' things." Well, in the real world discrete "things" come in nonnegative integer quantities ... or do they? For instance, we could make an agreement that "-3 things" means "I'm missing 3 things" or "you owe me 3 things". Nothing stops us from making agreements about the use of words. So maybe we could "agree" that "(3 + 4 i) times" means exactly an occurrence of the expression "3 + 4 i" in a context like this one. Is there any harm in that? :-)