Why the Square Root of 2 is Irrational | square root of 2
TheSquareRootof2Isthesquareroot[1]of2afraction?Letusassumethatitis,andseewhathappens.Ifitisafraction,thenwemustbeabletowriteitdownasasimplified[2]fractionlikethis:m/n(mandnarebothwholenumbers[3])Andwearehopingthatwhenwesquareitweget2:(m/n)2=2whichisthesameasm2/n2=2orputanotherway,m2istwiceasbigasn2:m2=2×n2HaveaTryYourselfSeeifyoucanfindavalueformandnthatworks!Example:letustrym=17andn=12:m/n=17/12Whenwesquarethatweget172/122=289/144=2.0069444...Whichiscloseto2,butnotquiterightYoucanseewereall...
The Square Root of 2Is the square root[1] of 2 a fraction?
Let us assume that it is, and see what happens.
If it is a fraction, then we must be able to write it down as a simplified[2] fraction like this:
m/n
(m and n are both whole numbers[3])
And we are hoping that when we square it we get 2:
(m/n)2 = 2
which is the same as
m2/n2 = 2
or put another way, m2 is twice as big as n2:
m2 = 2 × n2
Have a Try YourselfSee if you can find a value for m and n that works!
Example: let us try m=17 and n=12:
m/n = 17/12
When we square that we get
172/122 = 289/144 = 2.0069444...
Which is close to 2, but not quite right
You can see we really want m2 to be twice n2 (289 is...