When we multiply 0 by any other number we end up with 0.
C
Remember that the product of a number and its reciprocal is always 1.
A
Always
B
Sometimes
C
Sometimes
Practice makes perfect
We want to determine if the following sentence is always, sometimes, or never true.
A rational number multiplied by a rational number is rational.
Let's recall that each rational number can be written as a fraction where numerator and denominator are integers. Let ab and mn represent any two rational numbers where a, b, m, and n are integers. We will try to multiply them.
The product of two integers is also an integer. This means that the product of two rational numbers is a fraction where we have integers both in the numerator and the denominator. Therefore, a rational number multiplied by a rational number is always rational.
Let's take a look at the given sentence.
A rational number multiplied by an irrational number is rational.
We know that 0 is a rational number, and when we multiply 0 by any number we end with 0. This means that the product of 0 and an irrational number is a rational number.
0 * π & =0
0 * sqrt(2) & =0
However, let's think what happens if we multiply a rational number other than 0 by an irrational number.
8 * π & =8π
3 * sqrt(2) & =3sqrt(2)
We can see that this time we end up with an irrational number. Therefore, a rational number multiplied by an irrational number is sometimes rational.
Let's take a look at the third sentence.
An irrational number multiplied by an irrational number is rational.
If we multiply two totally different irrational numbers we end up with an irrational number.
sqrt(2) * π & =sqrt(2)π
sqrt(3) * sqrt(2) & =sqrt(6)
However, if we multiply an irrational square root by itself, we end up with a whole number. Also if we multiply any number by its reciprocal we always end up with 1.
sqrt(2) * sqrt(2) & = 2 [0.7em]
Ď€ * 1/Ď€ & =1
This means that it is possible to obtain a rational number after multiplying two irrational numbers, but only in the cases we described. Therefore, an irrational number multiplied by an irrational number is sometimes rational.
