Notational issues: 1/2 Pi (D/2)^2 + 1/2 Pi (d/2)^2 - 1/2 Pi ((D-d)/2)^2

[Simonsky]

A particular question has revealed to me that I can easily get confused by algebraic notion , particularly when fractions are involved in multiplying out brackets:

I need to simplify: 1/2 Pi (D/2)^2 + 1/2 Pi (d/2)^2 - 1/2 Pi ((D-d)/2)^2

Trying to do this, I get: (PiD^2)/8 + (Pid^2)/8 - (PiD^2 -Pi2Dd + Pid)/8

Hmmm......think I've messed that up...can someone check this out before I go one? Thanks

 

[Simonsky]

A particular question has revealed to me that I can easily get confused by algebraic notion , particularly when fractions are involved in multiplying out brackets:

I need to simplify: 1/2 Pi (D/2)^2 + 1/2 Pi (d/2)^2 - 1/2 Pi ((D-d)/2)^2

Trying to do this, I get: (PiD^2)/8 + (Pid^2)/8 - (PiD^2 -Pi2Dd + Pid)/8

Hmmm......think I've messed that up...can someone check this out before I go one? Thanks

That should be 'can someone check this out before I go ON!'

 

[mmm4444bot]

… I need to simplify: 1/2 Pi (D/2)^2 + 1/2 Pi (d/2)^2 - 1/2 Pi ((D-d)/2)^2

… I get: (Pi D^2)/8 + (Pi d^2)/8 - (Pi D^2 - Pi 2 D d + Pi d)/8 …

You're doing fine. One error (shown in red above). Fix that, then do the subtraction, to finish.


PS: Did I mention previously that it's better to type Pi*d instead of Pid, or Pi*2*D*d instead of Pi2Dd? There's a good reason for this.

 

[mmm4444bot]

For future readers, do you know? Changing signs on terms is subtraction. That is, we can subtract expressions in algebra by adding their opposite.

Here's a symbolic example: Simplify the following.

(A^2)/n + (C^2)/n - (A^2 - B + C^2)/n

1/n * [ A^2 + C^2 - (A^2 - B + C^2) ]

Working inside the square brackets, we need to subtract the trinomial A^2 - B + C^2 (we want to get rid of the parentheses, then we can combine like-terms). To subtract, we add the opposite, instead. We get the opposite via multiplication by -1.

1/n * [ A^2 + C^2 + (-1)(A^2 - B + C^2) ]

Using the Distributive Property, the sign changes on each trinomial term and we remove the parentheses.

1/n * [ A^2 + C^2 - A^2 + B - C^2 ]

We combine like-terms.

1/n * [ A^2 + C^2 - A^2 + B - C^2 ]

1/n * [ B ]

Answer: B/n

 

[Simonsky]

You're doing fine. One error (shown in red above). Fix that, then do the subtraction, to finish.


PS: Did I mention previously that it's better to type Pi*d instead of Pid, or Pi*2*D*d instead of Pi2Dd? There's a good reason for this.

Thanks for the useful advice especially about negative signs in front of brackets.

So I should have had: (PiD^2)/8 + (Pid^2)/8 - (PiD^2 -Pi2Dd + Pid^2)/8

Respelling: (Pi*D^2)/8 + (Pi*d^2)/8 - (Pi*D^2 - Pi*2Dd + Pi*d^2)/8

So I have to now bear in mind the negative sign in front of the bracket: 1/8(Pi*D^2 + Pi*d^2 - Pi*D^2 + Pi*2Dd -Pi*d^2)

The two Pi*D^2 cancel out to leave: Pi*d^2 + Pi*2Dd - Pi*d^2 which leaves (Pi*2Dd)/8 which I think is the same as: Pi*1/4Dd

Which can be re-written( I think) as 1/4PiDd

 

[mmm4444bot]

… Which can be re-written (I think) as 1/4PiDd

That's correct!

I would type it differently.

1/4*Pi*D*d

Pi*D*d/4

(Pi/4)(D*d)

What you typed would be problematic, when interacting with machines, and it could be misinterpreted by humans, too. :cool:

 

[mmm4444bot]

Thanks for the useful advice especially about negative signs in front of brackets.

You're welcome. After enough practice, you'll realize you can subtract positive terms as normal and only add the opposite when subtracting negative terms (versus distributing the negative across every term and adding everything). It works out the same, either way.

A^2 + C^2 - (A^2 - B + C^2)

When I look at the subtraction of what's inside the grouping symbols, I think of subtracting A^2 and C^2 and adding the opposite of -B:

A^2 + C^2 - A^2 + B - C^2

That is, I don't actually multiply the trinomial by -1 and then add the resulting "opposite" terms. In my head, I visualize subtracting terms that look positive and adding the opposite of terms that look negative (i.e., symbols that have either a negative sign or a subtraction operator in front of them).

-x - (-y + z - u)

I need to subtract what's in the grouping symbols. I'm going to subtract z (because it looks positive) and -y and -u look negative, so instead of subtracting them I'm going to add their opposite:

-x + y - z + u

---

I also want to make a note about "positive terms" and "negative terms". We need to be careful, when saying "positive" or "negative" because we don't know the actual numbers we're dealing with.

A, -B, C

Those are three terms, and we say "negative B" to m͏ean there's a negative sign in front of symbol B. We're not saying that symbol -B represents a negative number.

If a situation has B = 4 (or some other positive number), then -B represents -4 and -B is truly a negative number.

But in a different situation we might have B = -1/7 (or some other negative number). Then symbol -B represents 1/7 so it's not true that -B is a negative number.

Again, we write A - B + C and we say that A and C are positive and we say that B is negative (because there's a subtraction operator in front of symbol B), but that has more to do with the symbolism. We can't say anything about the sign of actual numbers until after we know what the numbers are.

-x

Is that a negative number? If that's all we're given, then we can't say.

If we know that symbol x itself represents a positive number, then we can say that -x represents a negative number.

If we know that symbol x itself represents a negative number, then we can say that -x represents a positive number.

-x could turn out to be neither positive nor negative (that happens when x=0).


Keep up your practice, and all of this will become second nature, in time. :cool:

 

[Steven G]

Thanks for the useful advice especially about negative signs in front of brackets.

So I should have had: (PiD^2)/8 + (Pid^2)/8 - (PiD^2 -Pi2Dd + Pid^2)/8

Respelling: (Pi*D^2)/8 + (Pi*d^2)/8 - (Pi*D^2 - Pi*2Dd + Pi*d^2)/8

So I have to now bear in mind the negative sign in front of the bracket: 1/8(Pi*D^2 + Pi*d^2 - Pi*D^2 + Pi*2Dd -Pi*d^2)

The two Pi*D^2 cancel out to leave: Pi*d^2 + Pi*2Dd - Pi*d^2 which leaves (Pi*2Dd)/8 which I think is the same as: Pi*1/4Dd

Which can be re-written( I think) as 1/4PiDd

Looks to me that you wrote 1/(4PiDd)