Hi there and thanks for joining me today.
In this miniature world in which we find
ourselves
the houses are constructed with all of
their proportions twelve times smaller
than usual. That can be written as a
ratio as 1 to 12. A one centimeter item
here would become 12 centimeters in the
"real" world. A ratio is a relation between
two numbers. The ancient Greeks called
ratios 'logos' which had a few meanings
including word and reason. When Greek
mathematical texts were translated into
Latin the word ratio was used. In Latin
ratio meant something that was reasoned
out, calculated or thought through. Our
good friend pi is a ratio. The ratio of a
circle's circumference to its diameter.
Pi cannot be expressed as the ratio of two
integers and so it is called irrational.
Isn't it funny that a ratio is also
irrational, how can that be?
Well circumference and diameter are not
necessarily integers. In fact it will
always be the case here that if the
diameter is an integer, the circumference
is not an integer and if the
circumference is an integer, the diameter
is not an integer, precisely because pi
is irrational.
The diagonal of any square can be found
by Pythagoras and using a square with
side length one we can see that the
ratio from side to the diagonal will
always be 1 to the square root of 2.
Square root of 2 is another irrational
number. It is rumoured that the Greek
Hippasus who first discovered the
existence of irrational numbers was
drowned at sea by his fellow
pythagorean's as they found his work too
shocking to accept.
Here we have a circle
inside of a square. As a little challenge
let's try to find the ratio of the area
of the circle to the area of the square.
To find the area of the circle we would
use pi * r squared where r is the radius
and since there are two radii in a side
we could square a side and get 4 * r
squared as the area of the square.
Cancel the r squared's and you get that the ratio
of the circle's area to the square's area
is pi to 4. This is always true for a
circle inscribed in a square.
And so that's been our little look at ratios,
thanks for watching.
