Explaining multiplication of fractions

Question

The best way I've been able to describe multiplication is as $$ a\times b = \sum^a_{i=1} b$$ But my definition does not account for things such as $2.99792458\times8.987551787$ and $\frac{1}{7}\times\frac{2}{9}$ When it comes to fractions and decimals, I can't seem to explain it intuitively.

According to Wikipedia, for fractions

Generalization to fractions $\frac{A}{B}\times \frac{C}{D}$ is by multiplying the numerators and denominators respectively: $\frac{A}{B}\times \frac{C}{D} = \frac{(A\times C)}{(B\times D)}$. This gives the area of a rectangle $\frac{A}{B}$ high and $\frac{C}{D}$ wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.

Is there a better more intuitive way of expressing the product of two fractions than something as vague as the area of a rectangle? (atleast it seems vague to me)

Answer 1

I will speak from a historical perspective in defense of multiplication as corresponding to area.

Human cultures did not develop a symbolic understanding of multiplication for quite some time (perhaps even as late as the ancient Greeks, though I may be off here). The earliest recorded bits of evidence demonstrating human knowledge of multiplication show that multiplication was first understood geometrically: the quantity that we would now call $a\times b$ was defined to be the area of the rectangle of height $a$ and width $b$ to all early civilizations that knew of the concept, including the Mesopotamians (incl. Babylonians) and Egyptians.

From this we might argue that the area of a rectangle is the most intuitive way to understand multiplication. The convenient aspect of this definition is that it is universal across all number systems that are a subset of the real numbers. As soon as one fixes a unit of length, there is a correspondence between any real number and a particular length (though the definition may be unwieldy for irrational numbers), and thus for any two real numbers $a$ and $b$ the rectangle of height $a$ and width $b$ is intuitively defined. In particular it is very easy to define geometrically the length corresponding to the fraction $a/b$, so this approach works very naturally for rational numbers, and indeed this is how ancient cultures understood multiplication of fractions. This universality in the definition is an advantage that the concept of integer multiplication as "repeated addition" lacks.

Later on, of course, humans developed the algebraic understanding of multiplication and generalized it to more abstract algebraic structures; from a certain point of view one could try to understand multiplication as this sort of algebraic formalism, but then again this is not so much "trying to understand" as "deciding not to bother."

Answer 2

If you accept, say, $3\times 7$ is $3$ groups of $7$, then you can move on to certain types of fractions times whole numbers. For instance $\frac23 \times 12$ is $\frac23$ of a group of $12$. Since one-third of $12$ is $4$, two-thirds of $12$ is two fours or $8$.

We could also do a problem like $\frac23 \times \frac{12}{17}$ by realizing that $\frac{12}{17}$ is $12$ items (each item is one-seventeenth of a whole). Since $\frac23$ of $12$ items is $8$ items, we get that $\frac23\times \frac{12}{17}=\frac8{17}$.

If the values aren't compatible, we can use equivalent fractions. For instance, let's say we want to do $\frac23\times \frac57$. We rewrite $\frac57$ so that the numerator can be taken by thirds. That is $$\frac23\times\frac57=\frac23\times \frac{15}{21}.$$ But $\frac13$ of fifteen twenty-firsts is five twenty-firsts. So two-thirds of fifteen twenty-firsts is two groups of five twenty-firsts, that is, ten twenty-firsts.

We've argued that $$\frac23\times\frac57=\frac{10}{21}.$$ Other fraction multiplication problems can be done in a like manner.

Answer 3

The historical reason why multiplication is defined the way it is comes from how multiplication used to be interpreted when it came to fractions. If you have a rectangular block of squares and there are $6$ squares across with $8$ square upwards you would say your rectangular block consists of $6\times 8$ squares. In a way multiplication is measuring "area" by unit squares.

Now if you introduce fractions you can still interpret multiplication as a sort of area. Let us say that horizontally there are $\tfrac{3}{2}$ squares and vertically there are $\tfrac{5}{3}$ squares. We want to find the area of such a square. To imagine this computation it is best to introduce common denominators and break up each unit side into $\tfrac{1}{6}$ of what it was. Now the horizontal side has $9$ sixth-squares and the vertical side has $10$ sixth-squares. In total there are $90$ sixth-squares, this is the area in terms of sixth-squares. In terms of unit squares this figure is then $\frac{90}{36} = \frac{15}{6} = \frac{5}{2}$ because each unit square contains $36$ sixth-squares.

This example shows that if you want to think of multiplication as counting the area then you have to define it by this rule for fractions.

Answer 4

$\frac ab \times \frac cd$=$\frac ab \times c \times \frac 1d$.

$\frac ab \times c \times \frac 1d$=$a \times \frac 1b\times c \times \frac 1d$

$a \times \frac 1b\times c \times \frac 1d$=$a \times c \times \frac 1b \times \frac 1d$.

$\frac 1b \times \frac 1d$=b$^{-1}$d$^{-1}$=(bd)$^{-1}$=[1/(bd)].

Thus, $a \times c \times \frac 1b \times \frac 1d$=ac[1/(bd)]=[(ac)/(bd)]