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I always thought multiplication meant repeated addition. Consider

$$4 \times 3 = 12.$$

This the same as

$$4 + 4 + 4 = 12.$$

Now consider

$$\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}. $$

If I use repeated addition approach, then I get

$$\frac{1}{2} + \frac{1}{2} = 1. $$

So the repeated addition approach doesn't really work.

I was helping my nephew with his maths homework and I was using the repeated addition approach. Then multiplying fractions came up and now I'm confused. Is there an easy answer to this?? Anything I could read. I'm very interested in trying to understand it. Right now it feels like I know nothing. lol

Dave L. Renfro
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troy beckett
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    Multiplication is more of a scaling or stretching operation than repeated addition. So when you multiply by $1/2$ you're really just shrinking everything down to half the size just as multiplying by $2$ doubles everything. – CyclotomicField Jan 12 '24 at 14:40
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    $\frac12\times\frac12$ would be addition of half copy of $\frac12$, not $2$ copies. – peterwhy Jan 12 '24 at 14:40
  • Multiplication is area. If you have a rectangle with four unit segments along along one side and three unit segments along another side, altogether it consists of twelve squares altogether. If you have a rectangle with half of a unit segment along one side and a half of a unit segment along another side, altogether it consists of a quarter of a square. – Lee Mosher Jan 12 '24 at 14:44
  • This comes down to what context (i.e. mathematical object) multiplication takes place in. For example, multiplication is certainly repeated addition when dealing with the positive integers, but not so much when dealing with $3 \times 3$ matrices with complex-number entries (or even when just dealing with the complex numbers). That said, as @peterwhy points out, your repeated addition example is not correctly carried out. – Dave L. Renfro Jan 12 '24 at 14:45
  • @peterwhy wow that's stupid of me lol should I delete the post. Can't believe that I've missed that – troy beckett Jan 12 '24 at 14:50
  • As for multiplying fractions, $\dfrac{a}{b}\times \dfrac{c}{d} = \dfrac{a\times c}{b\times d}$ where here the multiplication within the numerator is the multiplication you are used to... similarly for the denominator. We can then "simplify" the fractions as necessary, recalling what it means for two fractions to be equal. – JMoravitz Jan 12 '24 at 14:52
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    should I delete the post --- Personally, I think your error might be worth having available for others to see in the future. Maybe you can answer your question, but you'll need to use correct math formatting in both the question and answer. However, I suspect others might downvote your question anyway (and perhaps also an answer, if you give one?), so maybe it's not worth it. – Dave L. Renfro Jan 12 '24 at 14:55
  • @JMoravitz I've actually already tried to read that post but I found the answers to complex. – troy beckett Jan 12 '24 at 14:55
  • @DaveL.Renfro ok thanks. I'll look up correct formatting wasn't aware I was doing something wrong. – troy beckett Jan 12 '24 at 14:57
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    In your repeated addition scenario $\frac12 + \frac12$ is actually $2 \cdot \frac12$ which is why it equals $1$. – CyclotomicField Jan 12 '24 at 15:05
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    I formatted your post a bit. See How can I format mathematics here? and the links given there for reference. In your case, dollar signs go around the separate math equations/expressions and the multiplication symbol's code is \times (the +, =, - signs are those on your keyboard). – Dave L. Renfro Jan 12 '24 at 15:08
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    Maybe you'll find this analogy useful? – JonathanZ Jan 12 '24 at 15:11
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    This is a good question that already has answers. Related: https://math.stackexchange.com/questions/4187234/i-need-intuition-about-fraction-exponents-like-41-2-what-exactly-is-it/4187256#4187256 and https://math.stackexchange.com/questions/2214839/exactly-how-does-the-equation-nn-1-2-determine-the-number-of-pairs-of-a-given/2214863#2214863 – Ethan Bolker Jan 12 '24 at 15:12

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