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Intuitively, if addition can be interpreted as combining sets, then what can multiplication and division be understood as? A few more extensions to this:

  • What does multiplying and dividing by a decimal number mean intuitively? And why, when we multiply and divide by decimal numbers, do we need to move decimal points up and down based on what we are multiplying?

  • Why, when we divide by decimal numbers/fractions, do we reciprocal the fraction, then multiply by the denominator and divide by the numerator? Is there an intuitive explanation for this?

Can you try to keep the explanation as simple as possible? Because I'm still a beginner, if the explanation is too complex, I may not be able to understand it.

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  • $\begingroup$ If you're thinking of addition as corresponding to the disjoint union of two sets (I'm guessing this is what you mean by "combining" sets), then multiplication will correspond to the Cartesian product of two sets. $\endgroup$ – Greg Martin Jun 20 '18 at 6:11
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One good idea is to think of positive numbers as representing ratios, porportions, or scale factors. If we have an object, of any number of dimensions, and we strech or shrink it so that all distances between points in the object are multiplied by a constant number, the scale factor, this is called a Homothetic transformation. Notice that the scale factor is the constant ratio connecting the original distance between two points and the distance between the transformed points.

If we have two such transformation and we combine them together, one after the other, the result is another transformation whose scale factor is the product of the two scale factors. This result is the interpretation of multiplication of scale factors.

If we reverse such a transformation we get another transformation whose scale factor is the reciprocal of the original scale factor. This means we reverse the roles of before and after. In other words, we switch numerator with denominator.

Now if we apply one transformation and then apply the reverse of another, the result is another transformation whose scale factor is the product of the first scale factor and the reciprocal of the second. This result is the interpretation of division of scale factors.

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You can think of both multiplication and division in terms of sets; in fact, that's how many "operations" are defined in Abstract Algebra.

First of all, both multiplication and division can be thought of as "pairing" two elements.

Multiplying two numbers can be thought of counting the number of "pairs" that you can make out of two(or more) sets. For example, if you have $\{1,2,3\}$ and $\{1,2,3,4\}$, there would be 12 pairs: $(1,1), (1,2),..., (3,4)$.

Division can be thought of as "assigning" a number to each pair. When doing so, however, we identify some pairs as the same: For our usual Euclidean Division Algorithm, we identify two pairs as the same if they are "proportionate" to each other. For example, we identify the "pair" $(1,2)$ as $\frac{1}{2}$. Even though $(2,4)$ is a different "pair", since they are "proportionate", they become the same "pair".

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You have asked three or four essentially different questions in the same post. Please separate them from now on - then it's easier to get answers, and easier for others to use your questions and answers.

These are interesting questions each of which calls for a long answer. Here are brief responses.

The Greeks did their arithmetic mathematically as geometry. Adding was placing line segments end to end. Multiplication constructed rectangles. (I've no idea how they did everyday arithmetic for commerce.)

One contemporary way to think of arithmetic on the number line is to see "adding $x$" as a translation by $x$. That will move right when $x$ is positive and left when $x$ is negative, so deals with subtraction at the same time. Then "multiplying by $x$" is scaling everything by $x$. That's an expansion when $x > 1$ and a contraction when $0 < x < 1$. It reverses direction when $x$ is negative. This is the essence of the answer from @Somos .

The rule for "invert and multiply" is best understood when you think of a fraction like, say, $1/3$ as "what do you multiply $3$ by to get the answer $1$? That fits well with thinking of multiplication as scaling. It's more useful than the first elementary school discussion that models $1/3$ as dividing a pizza into three parts.

Numbers are just numbers. When you say "decimal numbers" you're just specifying how we write them in positional notation with base $10$. Then the rule that multiplying or dividing by a power of $10$ just moves the decimal point follows from the meaning of positional notation.

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