Primary Math?

[hazelm]

It's a very primary question, I know, but I got stuck in primary math "way back when".  Can someone please help?

 

Does anyone have a copy of "A Slice of Pi" by Liz Strachan?  Also a keyboard that will type what we see next to the hypotenuse of the triangle on page 9 - the square root takeoff of the division sign enclosing a squared + b squared.

 

Even if you do not have a keyboard that will type that equation, once you see it, can you please type how that is read aloud in English?  I know I am doubtless misreading it.  Books aren't always right but I rather suspect this one is.  It just isn't reading right to me. 

 

Thank you.

[Super Polymath]

This is pretty basic. Reverse to get Pythagorean theorem

 

 

hyp=√(a^2+b^2)

hyp^2=a^2+b^2

 

https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.549665.html

Edited June 18, 2017 by Super Polymath

[hazelm]

SuperPolymath, in all due respect, that is not what I am asking.  I understand quite well that the square of the hypotenuse is equal to the sum of the squares of the other two sides.   What I am asking is exactly how you would read that equation aloud in plain English.  I am asking because, the way I learned to read the square root symbol makes the sentence wrong.  Which, of course, means, I am reading it wrong.  I want to hear it right before I go further.  Thank you.

[DrKrettin]

I tried downloading it, but needed to register on something, so failed. I'm afraid I don't quite understand what you mean in the first post.

[hazelm]

Oh, dear.  I found the same line at Wiki and wanted to copy/paste it but it would not paste as it is written.  What it has is c = (radix symbol) a^2 + b^2

 

I did not want to say it as I learned it until I heard how a professional would read it.  If I am confusing then I might as well go ahead.  To me that says "c = the square root of a-squared plus b-squared.    Well, the square root of a-squared plus b-squared would be  a+b and that does not work.   In other words,  I learned that the radix says "square root of".  A friend tells me she learned it that way also.

 

Now you know why I want it typed as read aloud.  There is more:

 

By c, I assume it means c-squared but that isn't what it says either.  To me it just says "side c".  There is no ^2   beside it.

 

If I'm still confusing, I'll pass.

[exchemist]

Oh, dear.  I found the same line at Wiki and wanted to copy/paste it but it would not paste as it is written.  What it has is c = (radix symbol) a^2 + b^2

 

I did not want to say it as I learned it until I heard how a professional would read it.  If I am confusing then I might as well go ahead.  To me that says "c = the square root of a-squared plus b-squared.    Well, the square root of a-squared plus b-squared would be  a+b and that does not work.   In other words,  I learned that the radix says "square root of".  A friend tells me she learned it that way also.

 

Now you know why I want it typed as read aloud.  There is more:

 

By c, I assume it means c-squared but that isn't what it says either.  To me it just says "side c".  There is no ^2   beside it.

 

If I'm still confusing, I'll pass.

Perhaps your problem arises because √(a²+b²) is NOT (a+b ).   If you square (a+b ) you get a² + 2ab + b².  

 

From your description, c seems to be something I would express in spoken English as:  "the square root of a squared plus b squared", which I think is what you said also.   

Edited June 18, 2017 by exchemist

[hazelm]

Exactly Exchemist.  And our contradiction arises here:  "because √(a²+b²) is NOT (a+b )."  The difference is how we read that.  I  know you are right.  And I know why it looks wrong to me.   My problem comes with the symbol that encloses  the two sides squared.  It says, as I think you also said "the square root of".  Isn't the square root the base number?   So, wouldn't the square root of a-squared be a? 

 

There is a word for what I am doing - arguing against myself - and I apologize for that.  You and I put it in the same words but I am wrong in my conclusion and I can't see why.   I could go on with the triangle I have drawn to illustrate but it would get boring to you.

 

Let me think about it for a while.  Maybe it will come to me at 2:00 AM.  Thank you for the reading.  That does confirm what I thought it said.

 

P.S.  Suddenly, I saw this.  "If you square (a+b ) you get a² + 2ab + b²."  I didn't see that before but it is a "poser" for sure.  Thanks, I think.  :-)

Edited June 18, 2017 by hazelm

[DrKrettin]

Isn't the square root the base number?   So, wouldn't the square root of a-squared be a? 

 

 

 

Yes, the square root of (a squared) is (plus or minus) a

 

If c squared is a squared plus b squared, then square root © = square root (a squared plus b squared)

 

Edit: that's weird - I wanted to put c in brackets and it came up with that symbol

Edited June 18, 2017 by DrKrettin

[exchemist]

Exactly Exchemist.  And our contradiction arises here:  "because √(a²+b²) is NOT (a+b )."  The difference is how we read that.  I  know you are right.  And I know why it looks wrong to me.   My problem comes with the symbol that encloses  the two sides squared.  It says, as I think you also said "the square root of".  Isn't the square root the base number?   So, wouldn't the square root of a-squared be a? 

 

There is a word for what I am doing - arguing against myself - and I apologize for that.  You and I put it in the same words but I am wrong in my conclusion and I can't see why.   I could go on with the triangle I have drawn to illustrate but it would get boring to you.

 

Let me think about it for a while.  Maybe it will come to me at 2:00 AM.  Thank you for the reading.  That does confirm what I thought it said.

The square root of a² is a, certainly. But you obviously can't take a square root of a sum and expect it to be just the sum of the square roots of the individual components of the sum.

 

For example, 4+5 = 9. You can write √(4+5) = √9 = 3.

 

But if you try to take the roots of the two terms and add them, you get √4 + √5 = 2 +√5, and it is plain that √5 is NOT equal to 1. In fact it is a bit more than 2 !  So you cannot do that.

 

The square root of a² + b² is just " √(a² + b² ) " and you cannot reduce this any further, algebraically - you're just stuck with it written like that.  

Edited June 18, 2017 by exchemist

[DrKrettin]

And just to add to that (I've stolen exchemist's root sign) spoken language is ambiguous because you need to know where the brackets are, and they are annoyingly  silent.

 

If you say c squared = a squared + b squared 

 

then take the square roots of both sides, you get

 

c = √(a squared + b squared)

 

But when you say this, you could mean (but don't)

 

c = √(a squared) + b squared

[exchemist]

Exactly Exchemist.  And our contradiction arises here:  "because √(a²+b²) is NOT (a+b )."  The difference is how we read that.  I  know you are right.  And I know why it looks wrong to me.   My problem comes with the symbol that encloses  the two sides squared.  It says, as I think you also said "the square root of".  Isn't the square root the base number?   So, wouldn't the square root of a-squared be a? 

 

There is a word for what I am doing - arguing against myself - and I apologize for that.  You and I put it in the same words but I am wrong in my conclusion and I can't see why.   I could go on with the triangle I have drawn to illustrate but it would get boring to you.

 

Let me think about it for a while.  Maybe it will come to me at 2:00 AM.  Thank you for the reading.  That does confirm what I thought it said.

 

P.S.  Suddenly, I saw this.  "If you square (a+b ) you get a² + 2ab + b²."  I didn't see that before but it is a "poser" for sure.  Thanks, I think.  :-)

Just saw your PS. Yes, this is something in algebra that you need to know.

 

When you multiply two brackets together , like (a + b ) x ( c + d), you have to multiply each term in the second by each term in the first, so you get 4 terms, like this: ac + ad from multiplying by the first, PLUS  bc + bd, from multiplying by the second, i.e. ac + ad + bc + bd in total.

 

So, with (a+b )(a+b ) you get: a.a + a.b + b.a + b.b, in other words a² + 2ab + b².  

 

Again, to give a numerical example, let's look at (2+3)² = 5² = 25.  Applying the rule above to (2+3)(2+3) you would expand the brackets to get: 2.2 + 2.3 +3.2 + 3.3 = 4 + 6 + 6 + 9 = 25. Bingo!

 

It is a famous schoolboy error to think (a+b )² = a² + b² . If you try that in the numerical example above you get 2.2 + 3.3, which  gives you only  13. Fail! 

Edited June 18, 2017 by exchemist

[hazelm]

The square root of a² is a, certainly. But you obviously can't take a square root of a sum and expect it to be just the sum of the square roots of the individual components of the sum.

 

For example, 4+5 = 9. You can write √(4+5) = √9 = 3.

 

But if you try to take the roots of the two terms and add them, you get √4 + √5 = 2 +√5, and it is plain that √5 is NOT equal to 1. In fact it is a bit more than 2 !  So you cannot do that.

 

The square root of a² + b² is just " √(a² + b² ) " and you cannot reduce this any further, algebraically - you're just stuck with it written like that.  

Oh, I am remembering something similar that the author said in her book.  I am seeing the impossibility but forgetting to stop.  Hmmmm?  Thanks again.

[hazelm]

Just saw your PS. Yes, this is something in algebra that you need to know.

 

When you multiply two brackets together , like (a + b ) x ( c + d), you have to multiply each term in the second by each term in the first, so you get 4 terms, like this: ac + ad from multiplying by the first, PLUS  bc + bd, from multiplying by the second, i.e. ac + ad + bc + bd in total.

 

So, with (a+b )(a+b ) you get: a.a + a.b + b.a + b.b, in other words a² + 2ab + b².  

 

Again, to give a numerical example, let's look at (2+3)² = 5² = 25.  Applying the rule above to (2+3)(2+3) you would expand the brackets to get: 2.2 + 2.3 +3.2 + 3.3 = 4 + 6 + 6 + 9 = 25. Bingo!

 

It is a famous schoolboy error to think (a+b )² = a² + b² . If you try that in the numerical example above you get 2.2 + 3.3, which  gives you only  13. Fail! 

Hmmm?  What happened to BODMAS - do the brackets first? 

[DrKrettin]

Hmmm?  What happened to BODMAS - do the brackets first? 

 

Exactly - that's all he is doing - getting rid of the brackets.

[hazelm]

Exactly - that's all he is doing - getting rid of the brackets.

Without first adding the digits within them.  "The Times They ....." changed. 

[Turtle]

Seems to me that using Latex does not require parentheses to avoid ambiguity as does the ASCI symbol.[math]\sqrt{a^2+b^2}[/math]. Oui/no?

Edited June 18, 2017 by Turtle

[DrKrettin]

Without first adding the digits within them.  "The Times They ....." changed. 

 

This is the whole point of BODMAS - brackets first, addition much later.

 

Look - lets consider the number 5. If you square it, it's 25. OK?

 

Now look at a geometrical interpretation. A line of length 5 units. Square it, and you get a square of side length 5, area of 25.

 

Now look at that 5 as being 4 + 1

 

Squaring it gives (4 + 1) squared

 

Draw your square of side length 5. Mark the position of 4, along the top and down the side. Draw horizontal and vertical lines, and you have a square divided into 4 areas. These areas are 4 squared, 1 squared, a rectangle of 1 x 4 and another of 4 times 1

 

So 5 squared = (4 + 1)(4 + 1) = 4 squared + 1 squared + 4 + 4.....