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Why is $12\cdot 12=144$ but $8 \cdot 16=128$? a friend thinks it should be the same because $12+12=24$ and $8+16=24$. He said it is only $4$ that is moved to the other side. Please give some sort of reason so I can inform him. thanks

SQB
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christie
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    Take a piece of graph paper. Outline a rectangle $12$ squares wide and $12$ tall, and one rectangle $16$ by $8$. Have him count the number of squares in each of them. – Arthur Feb 03 '14 at 07:32
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    $2+2=3+1$, therefore $4=22=31=3$. – Michael Greinecker Feb 03 '14 at 07:33
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    I think your friend is trolling you... – Daniel Pietrobon Feb 03 '14 at 07:36
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    What does your friend think happens when "it is only 12 that is moved to the other side" and the product becomes $0\cdot 24$? – Blue Feb 03 '14 at 09:07
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    Tell your friend you will give him $8 \cdot 16$ dollars and the answer to his question if he gives you $12 \cdot 12$ dollars. – Andrej Bauer Feb 03 '14 at 09:11
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    Respond "Well, that's silly. There's absolutely no reason to believe that just because you move part of a number from one piece of a mathematical expression to another you'll get the same result. I mean, take $2^4$. If I take $2$ away from the exponent and add it to the base, I'll have $4^2$, and ... OH MY GOD!" – Blue Feb 03 '14 at 09:11
  • Indeed it is no coincidence that the difference is $4^2=16$. – Carsten S Feb 03 '14 at 09:19

3 Answers3

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Intuitive answer: Let's look at a simple case of $2\times2$ and $1\times3$. Those two definitely don't equal, in fact, the second will always be smaller than the first.

I highly recommend what Arthur suggested in the comments - in fact, you may even visualise why this is true by taking the bottom $4$ rows as collumns and cutting away the parts that "sticks out" - a $4\times4$ square (I would draw this, but my MathJax skills are sadly not up to par here.

Edit: Well I tried anyway. Take $4\times 4$ and $2\times 6$ \begin{matrix} \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ \color{red}{\cdot} & \color{red}{\cdot} &\color{red}{\cdot} &\color{red}{\cdot} \\ \color{red}{\cdot} & \color{red}{\cdot}&\color{red}{\cdot} & \color{red}{\cdot} \\ \end{matrix}

\begin{matrix} \cdot & \cdot & \cdot & \cdot & \color{red}{\cdot} & \color{red}{\cdot} \\ \cdot & \cdot & \cdot & \cdot & \color{red}{\cdot}&\color{red}{\cdot} \\ & & & & \color{fuchsia}{\cdot} & \color{fuchsia}{\cdot} \\ & & & & \color{fuchsia}{\cdot} & \color{fuchsia}{\cdot}\\ \end{matrix}

Mathematical answer: For $b >0$ $$ (a-b)(a+b)=a^2 - b^2 < a ^2$$

Dahn
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    Just to fill in the mathematical answer, this is what he means: $$8 \cdot 16 = (12 - 4)(12 + 4) = 12^2 - 4^2 < 12^2 = 144$$ – Arthur Feb 03 '14 at 07:39
  • Dammit, thanks for that. I've always been bad at inequalities ever since high school :> – Dahn Feb 03 '14 at 07:40
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    The first time I encountered inequality signs, in elementary school, I was told it symbolized the mascot of our textbook (a crow) gaping towards the biggest basket of apples, the greedy bastard. Has helped me since, though. – Arthur Feb 03 '14 at 07:45
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Suppose you have two numbers $x,y$ whose sum is $24$ and whose product is $144$. That is,

$$x+y=24,\\ x\,y=144.$$

Just knowing the sum and product, we can figure out what $x$ and $y$ must be. Subtracting $x$ from both sides in the first equation isolates $y$: $~~~~y=24-x.$ Substituting this expression for $y$ into the second equation yields:

$$x\,(24-x)=144\\ \Leftrightarrow 24x-x^2=144\\ \Leftrightarrow 0=x^2-24x+144=(x-12)^2\\ \Leftrightarrow x=12.$$

From $y=24-x$, we find $y=24-12=12.$

In general, a pair of numbers is determined by their sum and product. If you believed your friend who thinks that a pair's sum determines its product, then you end up with the following contradiction: $$\text{Suppose two different pairs of numbers have the same sum.}\\ \text{If they have the same sum, they also have the same product.}\\ \text{But if two pairs of numbers have both the same sum and product, the pairs are actually identical.}\\ \text{Since two things cannot be both different and the same, contradiction.}\\ $$

David H
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because $12+12=24$ and $8+16=24$.

Tell your friend that $0+24$ is also $24$... :-)

He said it is only $4$ that is moved to the other side.

Then tell him that the difference between $128$ and $144$ is also “only” $4^2=16$... :-)

Lucian
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