# Intuition behind multiplication

I recently read this post and the highest voted comment and it got me thinking. How does think about multiplication if it is decimals?

For example, if we have $3.9876542 \times 2.3156479$ then how would we multiply that? It doesn't make a lot of sense to add $3.9876542$, $2.3156479$ times. Then how would you think about multiplying that i.e. what's the intuition of behind that?

Thanks!

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You can write $5\cdot3$ or $5\times3$. There's no need for $5*3$. That is a device introduced for use when you're restricted to the characters on the keyboard and must use the letter x for something else. –  Michael Hardy Aug 10 '13 at 23:43

A rectangle with sides $3.9876542\,\mathrm m$ and $2.3156479\,\mathrm m$ can be viewed as $3987654200$ by $2315647900$ namometers instead. Then you can actually count all thos tiny square-nanometers (or simplify this by repaeted addition!) and obtain an area of $9234003074156180000\,\mathrm{nm}^2$. Since there are $1000000000000000000\,\mathrm{nm}^2$ in each $\mathrm m^2$, you end up with $9.23400307415618\,\mathrm m^2$ and thus we should have $3.9876542\cdot 2.3156479 = 9.23400307415618$.

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It's the area of a rectangle with side lengths $3.9876542$ and $2.3156479$.

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The term "decimal" is an abbreviation for decimal fraction.

And $3.9876542$ is a convenient abbreviation for $\dfrac{39876542}{10000000}$.

It is not difficult to give meaning to $\dfrac{a}{b} \cdot \frac{c}{d}$, where $a,b,c,d$ are positive integers. Area is a good choice. So is distance travelled if speed is $\frac{a}{b}$ and time is $\frac{c}{d}$.

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Just think in terms of fractions. With fractions, multiplication can be interpreted as meaning "of", like usual. For example, $\frac23 \times \frac57$ is $\frac23$ of $\frac57$ -- and if you didn't know how to multiply fractions, you could draw a picture or slice up a cake to figure out the answer.

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