Surds

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Stage 1 - Stage 3

Lesson

You may have heard the saying "One rotten apple spoils the whole barrel". Well, with irrational numbers, it's much the same idea. Once you have an irrational number in your calculation, you pretty much always end up with an irrational answer no matter what you do to it.

Let's try an example. What is the square root of $3$3? In other words, what number multiplies by itself to get to $3$3? From the chapter on "estimating roots", we know that it is somewhere between $1$1 and $2$2. If we use a calculator, we find that the answer is $1.7320508757$1.7320508757... . The ... means that the numbers keep going, and for irrational numbers like $\sqrt{3}$√3, they never become a nice simple pattern, but continue with what looks like a random string of numbers forever.

Now, if we take an endless irrational number like $1.7320508757$1.7320508757... and add a nice normal number like $2$2 to it, we get $2+\sqrt{3}=3.7320508757$2+√3=3.7320508757 ... . Because the numbers go on forever, and never repeat, this is still irrational.

$Rational+Irrational=Irrational$`R``a``t``i``o``n``a``l`+`I``r``r``a``t``i``o``n``a``l`=`I``r``r``a``t``i``o``n``a``l`

$Rational\times Irrational=Irrational$`R``a``t``i``o``n``a``l`×`I``r``r``a``t``i``o``n``a``l`=`I``r``r``a``t``i``o``n``a``l`

Is the result of the sum a rational or an irrational number?

$\frac{3}{11}+\sqrt{7}$311+√7

Rational

AIrrational

B

Is the result of the product a rational or an irrational number?

$\frac{3}{11}\times\sqrt{7}$311×√7

Rational

AIrrational

B

Watch out!

There is one important exception to the rule that multiplying a rational number and an irrational number gives an irrational number. That is when you multiply an irrational number by zero. Anything multiplied by zero is zero, which is a rational number, and that applies to numbers like $\sqrt{3}$√3 as well.

Now, what if we were to add or multiply two irrational numbers together? Say, $\sqrt{3}$√3, which is $1.7320508757$1.7320508757..., and $\sqrt{5}$√5, which is $2.2360679775$2.2360679775... . Without actual doing it, I'm sure you can see that adding or multiplying these numbers together will give you an endless string of numbers, rather than a number which ends after a few places or repeats itself in a pattern. This brings us to our next two rules:

$Irrational+Irrational=Irrational$`I``r``r``a``t``i``o``n``a``l`+`I``r``r``a``t``i``o``n``a``l`=`I``r``r``a``t``i``o``n``a``l`

$Irrational\times Irrational=Irrational$`I``r``r``a``t``i``o``n``a``l`×`I``r``r``a``t``i``o``n``a``l`=`I``r``r``a``t``i``o``n``a``l`

Did you know?

There are in fact exceptions to both of these rules. See if you can figure them out!

1.$Irrational+Irrational$`I``r``r``a``t``i``o``n``a``l`+`I``r``r``a``t``i``o``n``a``l` makes a rational. Hint: One is positive, and one is negative.

2. $Irrational\times Irrational$`I``r``r``a``t``i``o``n``a``l`×`I``r``r``a``t``i``o``n``a``l` makes a rational. Hint: Think about the definition of $\sqrt{3}$√3.