square roots of rationals

square roots of rationals

0.1

Illustrative examples

The

square roots

of the positive rational numbers are either

rational

or

irrational

algebraic numbers

of degree (

http://planetmath.org/DegreeOfAnAlgebraicNumber

) two.

Here we consider the much used standard form into which the irrational square roots of positive rational numbers have to be simplified in that, for example, one could easily compare the results gotten by different pupils.

Such forms as

6

7

and

1.8

are ordinarily not used as final forms of calculations, i.e. one should not leave a

fractional number

for the

radicand

.  One can without greater trouble convert these cases such that the only radicand is a

positive integer

(which is not

divisible

by a square of an

integer

greater than 1):

6

7

=

42

49

=

42

7

,

(1)

1.8

=

18

10

=

9

5

=

3

5

.

(2)

Both of these results are quite simple, consisting only of the

quotient

of two numbers, one of which is a square root of an integer and the other an integer.  But the latter result is not standard because of that the square root is in the

denominator

; this situation can be changed by multiplying the numerator and the denominator by the square root:

1.8

=

18

10

=

9

5

=

3

5

=

3

⁢

5

(

5

)

2

=

3

⁢

5

5

(3)

True, the last form of (3) isn’t as simple as in (2), and of course it could be obtained more directly by multiplying the numerator and the denominator of the original radicand

9

5

by 5 such that its denominator would be the

square number

25:

1.8

=

18

10

=

9

5

=

9

⋅

5

25

=

3

⁢

5

5

In some situations, one may however prefer the result of (2) (cf.

properties of regular tetrahedron

).  Such forms have, though, the drawback that inexperienced pupils may give such results as

6

2

or

5

5

, which are without any division.

0.2

General formula

Generally, the square root of any

positive

rational

m

n

(where

m

,

n

∈

ℤ

+

) is in the form

m

n

=

p

⁢

d

q

=

p

q

⁢

d

,

(4)

where

p

,

q

,

d

∈

ℤ

+

,

gcd

⁡

(

p

,

q

)

=

1

and

d

is

squarefree

.  If

gcd

⁡

(

m

,

n

)

=

1

,  one has

q

=

n

.  The result is justified via the intermediate form

m

⁢

n

n

2

.  The form (4) demonstrates, that the square roots of positive rationals belong always to a

real quadratic field

ℚ

⁢

(

d

)

or to

ℚ

.

All values of the square roots of positive rational numbers belong to the real field

ℚ

⁢

(

2

,

3

,

5

,

7

,

11

,

13

,

…

)

of

infinite

degree over

ℚ

.

Remark 1.

The square roots of negative rationals have the correspondent form (4)
where

d

now is a negative squarefree integer and (4) belongs to the

imaginary quadratic field

ℚ

⁢

(

d

)

.

Title

square roots of rationals

Canonical name

SquareRootsOfRationals

Date of creation

2013-03-22 18:30:28

Last modified on

2013-03-22 18:30:28

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Topic

Classification

msc 11A25

Classification

msc 12F05

Synonym

accurate square roots of rationals

Related topic

SquareFree

Related topic

Gcd

Related topic

SquareRootOfSquareRootBinomial

Related topic

NumberField

Generated on Fri Feb 9 15:15:01 2018 by

LaTeXML