A blog about Mathematics and Mathematics Education in Singapore, as well as Mathematics Education in general. Written for students, parents, educators and other stakeholders in Singapore, and around the world.
More information here. Check out my Education and Technology blog. Follow me on Twitter.
Sunday, April 12, 2015
[AM_20150412SQS] Perfect Squares Lurking Absurdly
Surds are expressions involving roots like
square roots. They are usually irrational numbers. If you try to put them into a ratio of integers, you are absurd! Many
students (and teachers?) are not sure of how to put square roots of numbers in
a simple form. The trick is to use square numbers or perfect squares. These are squares of whole numbers. For example:-
= 1 ´ 1 = 1 Ö1 = 1
= 2 ´ 2 = 4 Ö4 = 2
= 3 ´ 3 = 9 Ö9 = 3
= 4 ´ 4 = 16 Ö16 = 4
= 5 ´ 5 = 25 Ö25 = 5
A number like 4
can be represented by a real square
whose sides have length 2 units.
Note also that if you take the square root of a perfect square, you
always get a nice whole number.
How do you
deal with numbers that are not perfect squares? You factor out as many perfect squares as
possible. This would eventually lead to surds
with small numbers, which are more manageable.
Here are some examples:-
With this weapon in our hands, let us kick some butt.
Moral of the Story
perfect squares reduces your square roots to surds involving square roots of
prime numbers, which are easier to combine or cancel. This gives a short and sweet solution. In mathematics, always try to do things by
the cleanest way (if you can).