# Question #d5958

Sep 1, 2017

If you think about it, the product will always be smaller than either of the factors individually. This is because multiplying by a decimal less than 1 will necessarily produce a result that is less than the original value. So as both factors are less than 1 the product is some fraction of one of the original factors.

For example let's consider 0.5 and 0.1.

Multiplying gives

$0.5 \cdot 0.1 = 0.05$

So we either take half of 0.1 or a tenth of 0.5 but either way the product ends up being smaller than both factors because the product corresponds to some fraction of BOTH factors. This means the product will always be smaller than both factors as it can be expressed as the smaller factor divided by some number.

Sep 1, 2017

The product will always be less than both of them...

(but see explanation)

#### Explanation:

...if they are less than one, but greater than zero.

When you multiply two decimals that are greater than zero but less than one, you are calculating a fraction of a fraction.

If one or both of your input decimals are negative, then the strongest statement you can make is that the absolute value of the product is always less than the absolute value of either of them.

Sep 1, 2017

Expanding on a Proof

#### Explanation:

Let x and y be arbitrary real numbers between 0 and 1.

We start with $x < 1$.
Multiply both sides by y. Since $y > 0$, the inequality remains the same ("less than").
$x y < 1 y$
That is, $x y < y$.

Now go back to our other assumption: $y < 1$.
We do the same thing. Multiply both sides by x.
Since $x > 0$, the inequality remains the same ("less than").
$x y < 1 x$
That is, $x y < x$.

Therefore, $x y$ is less than both x and y.