Math Help - Surd Explanation

1. Surd Explanation

I have a rectangle sides of sq.rt 2 and 1

I cut it in half to show that the ratio of the new rectangles is teh same as the original.

Here's what I did:

New rectangle is sq.rt 2 / 2 : 1

That's as far as I got.

The book says:

sq.rt 2 / 2 : 1 = 1 / sq.rt 2 : 1 = sq.rt 2 : 1

-I don't understand how they do this last line.

Thanks.

2. Another problem:

Do these equate, if so how:

1 - 1 / sq.rt 2 = (sq.rt 2 - 1) / sq.rt 2

3. The first one.

So you cut the sqrt(2) by 1 rectangle into to equal parts. Each half is
(1/2)sqrt(2) by 1
Or, [sqrt(2)]/2 by 1
So the ratio of the new dimensions is
[sqrt(2)]/2 : 1

You want to check if that is the same as the ratio in the original, sqrt(2) : 1.

2 is sqrt(2)*sqrt(2), so,
[sqrt(2)]/2 : 1
= [sqrt(2)]/[sqrt(2) *sqrt(2)] : 1
= 1/[sqrt(2)] : 1
Multiply both by sqrt(2),
= 1 : sqrt(2) ---------------------same as the original.

--------------------------------------
The second one.

1 - 1 / sq.rt 2 = (sq.rt 2 - 1) / sq.rt 2

Clear the fractions, multiply both sides by sqrt(2),
sqrt(2) -1 = sqrt(2) -1

Hey, one step only.

------------------
Another way.

The RHS is (sqrt(2) -1) / sqrt(2)
So divide,
= sqrt(2)/sqrt(2) -1/sqrt(2)
= 1 - 1/sqrt(2)

That's it.

I have a rectangle sides of $\sqrt{2}$ and $1.$

I cut it in half to show that the ratio of the new rectangles is the same as the original.

Here's what I did:

New rectangle is $\frac{\sqrt{2}}{2}\,:\,1$ . . . . not quite correct

This is the original rectangle:
Code:
      * - - - - - - - - - *
|                   |
|                   |
1 |                   | 1
|                   |
|                   |
* - - - - - - - - - *
√2
The ratio of length-to-width is: . ${\color{blue}\sqrt{2}\,:\,1}$

Bisect it:
Code:
      * - - - - * - - - - *
|         |         |
|         |         |
1 |         |         | 1
|         |         |
|         |         |
* - - - - * - - - - *
½√2       ½√2

Each small rectangle looks like this:
Code:
      * - - - - - *
|           |
|           | ½√2
|           |
* - - - - - *
1
The ratio of length-to-width is: . ${\color{blue}1\,:\,\frac{\sqrt{2}}{2}}$

And we must show that the two ratios are equal.

The first ratio is: . $\frac{\sqrt{2}}{1} \;=\;{\color{red}\sqrt{2}}$

The second ratio is: . $\frac{1}{\frac{\sqrt{2}}{2}} \:=\:\frac{2}{\sqrt{2}}$

. . Rationalize: . $\frac{2}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}} \;=\;\frac{2\sqrt{2}}{2} \;=\;{\color{red}\sqrt{2}}$

Therefore, the ratios are equal.

5. 1 - 1 / sq.rt 2 = (sq.rt 2 - 1) / sq.rt 2

Sorry, what I meant to ask is how do I get from

1 - 1 / sq.rt 2

to

(sq.rt 2 - 1) / sq.rt 2

6. Originally Posted by Soroban

And we must show that the two ratios are equal.

The first ratio is: . $\frac{\sqrt{2}}{1} \;=\;{\color{red}\sqrt{2}}$

The second ratio is: . $\frac{1}{\frac{\sqrt{2}}{2}} \:=\:\frac{2}{\sqrt{2}}$

. . Rationalize: . $\frac{2}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}} \;=\;\frac{2\sqrt{2}}{2} \;=\;{\color{red}\sqrt{2}}$

Therefore, the ratios are equal.

I don't understand why in the first ratio you have used the 1 as the denominator and then as the numerator for the second ratio.

I don't understand why in the first ratio you have used the 1 as the denominator
and then as the numerator for the second ratio.
We can't order a proportion randomly.

In both ratios, it is Length-to-Width.

. . That is, .the longer side : the shorter side.

If you don't "line them up" correctly, you're asking for trouble.
Code:
      * - - - - - - - - - *
|                   |       * - - *
|                   |       |     |
40 |                   |     3 |     |
|                   |       |     |
* - - - - - - - - - *       * - - *
60                     2

These two rectangles are similar; the side are proportional.

But not because: . $\frac{60}{40} \:=\:\frac{2}{3}$ . . . . which is not true.

Get the idea?

8. Originally Posted by Soroban

We can't order a proportion randomly.

In both ratios, it is Length-to-Width.

. . That is, .the longer side : the shorter side.

If you don't "line them up" correctly, you're asking for trouble.
Code:
      * - - - - - - - - - *
|                   |       * - - *
|                   |       |     |
40 |                   |     3 |     |
|                   |       |     |
* - - - - - - - - - *       * - - *
60                     2

These two rectangles are similar; the side are proportional.

But not because: . $\frac{60}{40} \:=\:\frac{2}{3}$ . . . . which is not true.

Get the idea?

Yes. So in your example it should be 60/40 : 3/2

In the original example it's 1 over the second time because that's the longer side.

Thank you!

1 - 1 / sq.rt 2 = (sq.rt 2 - 1) / sq.rt 2

Sorry, what I meant to ask is how do I get from

1 - 1 / sq.rt 2

to

(sq.rt 2 - 1) / sq.rt 2

.........

1 - 1 / sq.rt 2 = (sq.rt 2 - 1) / sq.rt 2

Sorry, what I meant to ask is how do I get from

1 - 1 / sq.rt 2

to

(sq.rt 2 - 1) / sq.rt 2
1 - 1/sqrt(2)

= 1/1 -1/sqrt(2)

Combine them. The common denominator is sqrt(2),
= (1*sqrt(2) -1) / sqrt(2)
= (sqrt(2) -1) / sqrt(2)

11. Originally Posted by ticbol
1 - 1/sqrt(2)

= 1/1 -1/sqrt(2)

Combine them. The common denominator is sqrt(2),
= (1*sqrt(2) -1) / sqrt(2)
= (sqrt(2) -1) / sqrt(2)
I still don't get it.

(1 ) - 1 / sq.rt 2 is what I start with

Which is teh same as:

(1 / 1) - (1/ sq.rt 2) I got that.

Combining them would give: (1 - 1) / (1 - sq.rt 2)

You seem to be multiplying the (1 - 1) by sq.rt 2.?

I still don't get it.

(1 ) - 1 / sq.rt 2 is what I start with

Which is teh same as:

(1 / 1) - (1/ sq.rt 2) I got that.

Combining them would give: (1 - 1) / (1 - sq.rt 2) <----this is not correct.

You seem to be multiplying the (1 - 1) by sq.rt 2.?
You seem to be multiplying the (1 - 1) by sq.rt 2.?

No. I multiplied the numerator 1 of the 1/1 by sqrt(2).

It is subtraction of fractions.
Example, 1/2 -3/4
Common denominator is 4.
4 divided by 2 ...[equals 2], times 1, equals 2
4 divided by 4 ...[equals 1] , times 3, equals 3
So,
1/2 -3/4
= (2 -3)/4
= -1/4

Now compare that to your surd above.

13. Originally Posted by ticbol
You seem to be multiplying the (1 - 1) by sq.rt 2.?

No. I multiplied the numerator 1 of the 1/1 by sqrt(2).

It is subtraction of fractions.
Example, 1/2 -3/4
Common denominator is 4.
4 divided by 2 ...[equals 2], times 1, equals 2
4 divided by 4 ...[equals 1] , times 3, equals 3
So,
1/2 -3/4
= (2 -3)/4
= -1/4

Now compare that to your surd above.
Ok:

(1 / 1) - (1/ sq.rt 2)

common denomintaor is sq.rt 2

So,

sq.rt 2 divide by 1 = sqrt2/1 times 1 is sq.rt2/1
sq.rt 2 divide by sq.rt 2 = 1 times 1 is 1

sq.rt 2 - 1 / sq.rt 2

Voila!!

I didn't know that subtraction method for fractions! Thanks. I know it now.