Question

Can I write `sqrt2` without writing `sqrt2` or `2^(1/2)`? If yes, how?

Answer 1

See below:

Explanation:

Mathematically speaking, I don't think that there is any other way to designate `sqrt2` without using

`sqrt2 or 2^(1/2)`

However, you could manually write it out as:

`"The square root of 2"`

Answer 2

Shown below

Explanation:

Im not entirely sure what you're looking for, but here's an idea

Use log laws:

`e^lnx = x , AA x in RR`

`=> e^ln(2^(1/2) ) = 2^(1/2)`

Another law: `alpha log beta = log beta^alpha`

`=> color(red)(e^( 1/2 ln2) ) = 2^(1/2)`

Another idea:

`sqrt(2) = ( sqrt(3)*sqrt(2) ) / sqrt(3) = sqrt(6)/sqrt(3)`

or in general `= sqrt(2n) / sqrt(n)`

Answer 3

`[1;bar(2)]`

Explanation:

Any positive square root of an integer can be expressed as a repeating continued fraction.

For example, suppose for some `x > 0`:

`x = 1+1/(1+x)`

Multiplying both sides by `(1+x)` this becomes:

`x^2+x = x+1+1`

Subtracting `x` from both sides, this becomes:

`x^2 = 2`

So we have found:

`sqrt(2) = 1+1/(1+sqrt(2))`

`color(white)(sqrt(2)) = 1+1/(2+1/(2+1/(2+1/(2+...))))`

One notation for continued fractions uses square brackets, a semicolon to separate the integer part from the coefficients in the fraction and commas to separate the coefficients.

So we could write:

`sqrt(2) = 1+1/(2+1/(2+1/(2+1/(2+...)))) = [1;2,2,2,2,...] = [1;bar(2)]`

Note the use of the viniculum (over bar) to indicate the terms that repeat.

Answer 4

`((0, 1), (2, 0))`

Explanation:

If you are familiar with matrix arithmetic then note first that matrices of the form:

`((a, 0), (0, a))`

where `a in QQ` (the set of rational numbers) are closed under addition, multiplication, additive inverse and reciprocal of non-zero matrices.

In fact the set of such matrices is isomorphic to the rational numbers `QQ` with the normal definitions of addition and multiplication, via the mapping `a -> ((a, 0), (0, a))`

Next consider the matrix `M = ((0, 1), (2, 0))`

We find:

`M^2 = ((0, 1),(2, 0))((0, 1),(2, 0)) = ((2, 0),(0, 2))`

In other words, `M` is a square root of `2`.

Hence we can map any number of the form `a+bsqrt(2)` where `a, b in QQ` to the matrix `((a, b), (2b, a))`

The set of matrices of the form `((a, b), (2b, a))` where `a, b in QQ` under matrix addition and multiplication form a field isomorphic to the set of real numbers of the form `a+bsqrt(2)` under normal addition and multiplication.

So in a sense we can say `sqrt(2) = ((0, 1), (2, 0))`