What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
I think it&#x27;s okay to say: &quot;It&#x27;s useful to <i>visualize</i> (or <i>think of</i>) multiplication as repeated addition, in contexts where that makes sense.&quot; Properly qualified the statement seems true and unbojectionable, and I think it&#x27;s a useful way to help students understand multiplication. (It&#x27;s not the only way.)<p>The reservation I&#x27;d have with the statement &quot;Multiplication <i>is</i> repeated addition&quot; is that the use of the word &quot;is&quot; implies an <i>identification</i>, or a <i>necessary</i> derivation of one from the other. The operations aren&#x27;t identical, and one is not <i>necessarily</i> derived from the other. As operations, they&#x27;re distinct.<p>(The issue also has nothing per se to do with units. When you introduce units of length or area, you&#x27;re <i>using</i> numbers in a <i>particular applied context</i>. Outside of that context, numbers don&#x27;t have units.)<p>In <i>everyday usage</i> &quot;addition&quot; and &quot;multiplication&quot; are conventional designations. Conventions always have ambiguities and edge cases.<p>The most common <i>formal structure</i> in which you have both addition and multiplication is in a <i>ring</i>. A ring is a set with two operations, which we <i>call</i> &quot;addition&quot; and &quot;multiplication&quot; --- but those are just names. We could just as well have used &quot;foo&quot; and &quot;bar&quot;. We tend to use the standard addition and multiplication symbols for those two operations, but that is also just a convention&#x2F;convenience. (Note also that those operations are not unique.)<p>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.<p>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 &quot;0&quot;), every element has an additive inverse, and addition is commutative. That&#x27;s addition.<p>Then you say multiplication is associative (and nowadays, since people find it convenient to assume this) and multiplication has an identity element (conventionally denoted &quot;1&quot;).<p>At the moment, you have two independently defined operations, which have nothing to do with one another. &quot;Multiplication&quot; is therefore <i>not identical to</i> (or <i>defined as</i>) repeated addition. But it&#x27;s not too useful to have two independent unrelated operations. You <i>connect</i> the two operations by introducting the distributive axiom: For all a, b, c in your ring,<p><pre><code>  a * (b + c) = a * b + a * c   and   (a + b) * c = a * c + b * c.
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(scythe pointed out the importance of the distributive law in another reply.) The distributve axiom is huge! Consider<p><pre><code>  a * (b + c) = a * b + a * c.
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From left to right, it says that &quot;you can multiply out&quot;; from right to left, it says that &quot;you can take out a common factor&quot;, which are both standard operations in algebra.<p><i>Once the distributive property is available to connect the operations</i>, it explains why you can &quot;think of&quot; multiplication as repeated addition: For instance,<p><pre><code>  3 * 2 = 3 * (1 + 1) = 3 * 1 + 3 * 1 = 3 + 3.
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So you say &quot;3 times 2&quot; <i>can be thought of</i> as &quot;3 added to itself 2 times&quot;. It <i>does not</i> say that multiplication <i>is</i> repeated addition, if &quot;is&quot; means &quot;defined as&quot; or &quot;derived from&quot;,<p>Or consider one of the standard illustrations given to kids in grade school:<p><pre><code>  3 * 2 is   $ $ $   which is   [ $ $ $ ]
             $ $ $              [ $ $ $ ]
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This is 2 groups of 3, i.e. 3 + 3. So 3 * 2 = 3 + 3.<p>But what happened here? We relied on our physical intuition to &quot;know&quot; that putting the original 6 dollars into 2 bags in the second step didn&#x27;t change the number of dollars. But formally, it is<p><pre><code>  3 * 2 = 3 * (1 + 1) = 3 * 1 + 3 * 1,
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which is the distributive law again.<p>(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 &quot;the same as&quot; 2 groups of 3.)<p>(Someone might suggest from the dollars example that multiplication might have arisen historically as a shorthand for repeated addition. I don&#x27;t know math history well enough to say, but historical derivation doesn&#x27;t imply identity or logical derivation.)<p>So I think saying &quot;Multiplication <i>is</i> repeated addition&quot; is a little sloppy in the use of the word &quot;is&quot;, 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!)<p>Here&#x27;s something to think about. (I don&#x27;t have an opinion myself.) Suppose we have a complex multiplication: (7 i)(3 + 4 i). Following the interpretation above, I describe this as &quot;7 i added to itself (3 + 4 i) times&quot;. If you don&#x27;t like the sound of that ... why? You might say &quot;You can&#x27;t have &#x27;3 + 4 i&#x27; things.&quot; Well, in the real world discrete &quot;things&quot; come in nonnegative integer quantities ... or do they? For instance, we could make an agreement that &quot;-3 things&quot; means &quot;I&#x27;m missing 3 things&quot; or &quot;you owe me 3 things&quot;. Nothing stops us from making agreements about the use of words. So maybe we could &quot;agree&quot; that &quot;(3 + 4 i) times&quot; means exactly an occurrence of the expression &quot;3 + 4 i&quot; in a context like this one. Is there any harm in that? :-)