Mathematical notation question-JR High

[Wits' End]

New question at the end of this post (not thread, this post )
My son asked me a question from his math work today. According to the lesson I answered it incorrectly. After reading through the lesson I understand my error, however I want to know if the methods, means of notation are standard or peculiar to this curriculum (Math U See).
[Note-I wrote this post in Word with superscript 2's, this didn't translate to this post. Rather than hunting for the correct ASCII code or whatever, I'll leave it as 'translated'. So any 2 after a 5 is supposed to be a superscript--a 'raised to the second power symbol'. I know there is something to do with a "^" but i don't recall :thinking:]
So -52 = -25 is the problem and answer from his book. I take this as –5 x –5, which should be 25 (maybe +25 for clarity). The instruction takes this as –(52) so –(5x5) and –(25) ending with the answer –25.

My difference in thought is that I think –5 = (-5). The text indicates that –5 = (-)(5). I could sort of understand doing the operation, squaring, before applying the –1, multiplying, however the text doesn't indicate or imply reading a (–1)(52).

So is what I describe, standard, unique or somewhere in between?

New Question
Rounding numbers up to the nearest factor of 10. If everyone on earth (1*10^10), was able to count all the atoms in the universe (1*10^81), in a second for a year 1*10^8. If they chose to say a digit in the largest known prime number (1*10^13000000) it would take (1*10^12999900) years
That just seems like too large a number. Where did I go wrong other than my rounding numbers up, a lot?

[FONT=&quot]TIA[/FONT]

 

[Marduke]

My son asked me a question from his math work today. According to the lesson I answered it incorrectly. After reading through the lesson I understand my error, however I want to know if the methods, means of notation are standard or peculiar to this curriculum (Math U See).
[Note-I wrote this post in Word with superscript 2's, this didn't translate to this post. Rather than hunting for the correct ASCII code or whatever, I'll leave it as 'translated'. So any 2 after a 5 is supposed to be a superscript--a 'raised to the second power symbol'. I know there is something to do with a "^" but i don't recall :thinking:]
So -52 = -25 is the problem and answer from his book. I take this as –5 x –5, which should be 25 (maybe +25 for clarity). The instruction takes this as –(52) so –(5x5) and –(25) ending with the answer –25.

My difference in thought is that I think –5 = (-5). The text indicates that –5 = (-)(5). I could sort of understand doing the operation, squaring, before applying the –1, multiplying, however the text doesn't indicate or imply reading a (–1)(52).

So is what I describe, standard, unique or somewhere in between?

[FONT=&quot]TIA[/FONT]

translated with correct notation:

So -5^2 = -25 is the problem and answer from his book. I take this as –5 x –5, which should be 25 (maybe +25 for clarity). The instruction takes this as –(5^2) so –(5x5) and –(25) ending with the answer –25.

My difference in thought is that I think –5 = (-5). The text indicates that –5 = (-)(5). I could sort of understand doing the operation, squaring, before applying the –1, multiplying, however the text doesn't indicate or imply reading a (–1)(5^2).

But to answer your original question, it's a difference of order of operations.

-5^2 is assumed to be -(5^2) = -25. When squaring a negative, you must use parenthesis to make sure the square encompasses all values such as (-5)^2 = 25

Order of operations goes by "PEMDAS", which is an acronym for "Please Excuse My Dear Aunt Sally" which stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction"

In your case above, the exponent takes precedence BEFORE the -1 is multiplied.

 

[oronocova]

I'm no math major so someone who is can correct me. But as far as I understand it -5^2 (negative five squared) is the same as saying -5*-5 (negative five times negative five) which yeild a positive result: 25.
If the problem was -(5^2) which is to say "negative the square of 5", then the resul would be -25.
You are supposed to operate exponents before multiplications so it sounds like the book is treating the negative sign infront of a number as a -1 times that number. Such as in: -1*5^2, or "negative one times five squared." This is not how I remember learning math... odd.

edit I just read something on mathforums.org which helped make sense, here's how i take it....

I guess I would treat -5 as a number (a negative number) not an operation on the positive number 5. Another case would be -x in which case I would treat it as a negative operation on the variable x. I guess it depends on whether you treat -5 as the number
"negative five" or as "the negative operation on the postive number five".

 

[asdalton]

The order of operations is standard and should be taught the same everywhere. Computer code (including Microsoft Excel formulas) also follows these rules. Update: Not Excel! See the posts below.

Exponents are evaluated prior to multiplication and division, which in turn are evaluated before addition and subtraction. Operations of equal priority are evaluated left to right. Parentheses are needed to override the default order.

For a negative sign, you can think of this either as a subtraction from 0, or a multiplication by -1. Either way, the exponent is evaluated first.

5² = 25
-5² = -25
(-5)² = 25

The only time you can raise a negative number to a power without parentheses is when using a variable.

Example:
If x = -5,
then x² = (-5)² = 25,
and -x² = -(-5)² = -25

 

[Wits' End]

Red by me, green and underline added for reference

....

-5^2 is assumed to be -(5^2) = -25.[That is my question! Is it?] When squaring a negative, you must use parenthesis to make sure the square encompasses all values such as (-5)^2 = 25
In your case above, the exponent takes precedence BEFORE the -1 is multiplied.

I'm no math major so someone who is can correct me. But as far as I understand it -5^2 (negative five squared) is the same as saying -5*-5 (negative five times negative five) which yeild a positive result: 25. [Dittos for me]
If the problem was -(5^2) which is to say "negative the square of 5", then the resul would be -25.
You are supposed to operate exponents before multiplications so it sounds like the book is treating the negative sign infront of a number as a -1 times that number. Such as in: -1*5^2, or "negative one times five squared." This is not how I remember learning math... odd. [both 5 and 25 as well as -5 and -25 are odd, for that matter so is -1 how odd]

edit I just read something on mathforums.org which helped make sense, here's how i take it....

I guess I would treat -5 as a number (a negative number) not an operation on the positive number 5. Another case would be -x in which case I would treat it as a negative operation on the variable x. I guess it depends on whether you treat -5 as the number
"negative five" or as "the negative operation on the postive number five". [that is what it boils down to, I think]

 

[IlluminatingBikr]

From what I know... -5^2 is always assumed to be -(5^2) [which comes out to -25] unless it's explicitly written (-5)^2 [in which case the answer would be +25].

The exponent only operates on the number right next to it and it does not operate on the number along with it's sign. I think this has to do with being able to see "negative signs" as both negative signs and subtraction operators.

Take the example:
2^2 + x^2 = 4^2

To solve you'd subtract 2^2 from both sides which yields:
x^2 = 4^2 - 2^2.

In this case, you are subtracting the value of two squared and not squaring negative two. If the order of operations was different, you'd be required to write the equation like this:
x^2 = 4^2 -(2^2). So there's definitely a trade off with having the standard be what it is, but it allows us to not write parentheses half the time.

 

[Wits' End]

From what I know... -5^2 is always assumed to be -(5^2) [which comes out to -25] unless it's explicitly written (-5)^2 [in which case the answer would be +25].

The exponent only operates on the number right next to it and it does not operate on the number along with it's sign. I think this has to do with being able to see "negative signs" as both negative signs and subtraction operators.

Take the example:
2^2 + x^2 = 4^2

To solve you'd subtract 2^2 from both sides which yields:
x^2 = 4^2 - 2^2.

In this case, you are subtracting the value of two squared and not squaring negative two. If the order of operations was different, you'd be required to write the equation like this:
x^2 = 4^2 -(2^2). So there's definitely a trade off with having the standard be what it is, but it allows us to not write parentheses half the time.

What if you write the equation (x^2 = 4^2 - 2^2) as x^2 = 4^2 + -2^2? To my eyes it looks like you are adding 4 rather than subtracting. It seems much clearer to write x^2 = 4^2 + -(2^2). OK I'm done I think, I sent a request to a math friend that is a member here to put his bit in. However I think I'll admit defeat now :shrug: . Just one of those math things, how 2^(-2)

 

[Cyclops942]

I was a math major in college (in addition to being a computer science major), and both majors taught me the rule that corresponds to the PEMDAS acronym described above. They also both taught me that -5^2 should be interpreted as (-5)^2; the negative sign is and should always be intrepreted as the value of the number, unless parentheses were used to override this notational convention.

Just my $.02.

 

[greenLED]

I was ... taught me that -5^2 should be interpreted as (-5)^2; the negative sign is and should always be intrepreted as the value of the number, unless parentheses were used to override this notational convention.

That's how it was ingrained into me as well (many, maaaaaaaany moons ago).

 

[Wits' End]

I was a math major in college (in addition to being a computer science major), and both majors taught me the rule that corresponds to the PEMDAS acronym described above. They also both taught me that -5^2 should be interpreted as (-5)^2; the negative sign is and should always be intrepreted as the value of the number, unless parentheses were used to override this notational convention.

Just my $.02.

That's how it was ingrained into me as well (many, maaaaaaaany moons ago).

:twothumbs 2 4 me!

 

[asdalton]

I was a math major in college (in addition to being a computer science major), and both majors taught me the rule that corresponds to the PEMDAS acronym described above. They also both taught me that -5^2 should be interpreted as (-5)^2; the negative sign is and should always be intrepreted as the value of the number, unless parentheses were used to override this notational convention.

The problem is that this convention requires making a distinction between the negation sign (as in "-5") and the subtraction sign (as in "2 - 5"). This can be done in a textbook. However, computer languages use the same symbol for both, and it's impossible to distinguish the two in handwritten equations. And we never interpret a variable expression like -x² as the square of -x.

I checked and discovered that Excel treats a lone "-" as part of the following number, so that the expression =-5^2 returns the value 25. However, the expression =0-5^2 returns -25. This is confusing, and actually incorrect (the first one) according to what I learned.

Interestingly, Visual Basic in Excel interprets a = -5^2 as -25.

 

[2xTrinity]

I just typed "-5^2" into google, which promptly performed the following calculation:

[SIZE=+1]-(5^2) = -25[/SIZE]

The problem is that this convention requires making a distinction between the negation sign (as in "-5") and the subtraction sign (as in "2 - 5"). This can be done in a textbook. However, computer languages use the same symbol for both, and it's impossible to distinguish the two in handwritten equations. And we never interpret a variable expression like -x² as the square of -x.

Agreed there would be no reason to explicitly define an equation like (-x)^2 as any even exponent will produce a positive answer, anyway. Likewise, (-x)^3 is also pointless to use, as it will produce the same answer as -(x^3) every time. If we're going to talk programming, the latter actually requires considerably less CPU power to accomplish as well.

In most computer languages, this order of operations ambiguity will never come up, as if I wanted to program the equation y = -x^2, I would use:

y = -x*x

If I wanted to raise -x to some arbitrary exponent, as in y= -x^z, most computer languages would actually require a function call to accomplish this, which will have non-ambiguous notation such as:

y = -pow(x,z) or
y = pow(-x,z)

If I wanted to hard code "y = -5^2", I would have to write:

y = -25

 

[MicroE]

Google gives the answer as -25
Yahoo gives the answer as +25.

Amusing!

 

[Wits' End]

... Likewise, (-x)^3 is also pointless to use, as it will produce the same answer as -(x^3) every time. ....

uhm (-5)^3= -125 -(5^3)= +125 ('+' added for clarity and differentiation ) :thinking: I think

Google gives the answer as -25
Yahoo gives the answer as +25.

Amusing!

Well Google has just gone down in my estimation

 

[asdalton]

uhm (-5)^3= -125 -(5^3)= +125 ('+' added for clarity and differentiation ) :thinking: I think

(-5)^3 = (-5) x (-5) x (-5) = 25 x (-5) = -125

Odd exponents always work this way: (-x)^n = -(x^n).

In fact, there is a whole category of odd functions that work this way. For example, sin(-x) = -sin(x).

Even functions give the same result for positive or negative input:
(-x)^2 = x^2
cos(-x) = cos(x)

 

[greenLED]

:twothumbs 2 4 me!

That may just mean the 3 of us are wrong.

 

[Marduke]

The "PEMDAS" order of operations states that exponents comes before multiplication, which means 5^2 is computed BEFORE -1*5

The examples listed on Wiki agree (under "Examples from arithmetic")
http://en.wikipedia.org/wiki/Order_of_operations

edit I just read something on mathforums.org which helped make sense, here's how i take it....

I guess I would treat -5 as a number (a negative number) not an operation on the positive number 5. Another case would be -x in which case I would treat it as a negative operation on the variable x. I guess it depends on whether you treat -5 as the number
"negative five" or as "the negative operation on the postive number five".

Really? There is a section that has exactly the question asked here, and it follows PEMDAS. It again explains that exponents must be dealt with BEFORE multiplication.
http://mathforum.org/library/drmath/view/57375.html

 

[asdalton]

Here are a couple of additional links:

http://aah.ryan-usa.com/node27.html

http://mathforum.org/library/drmath/view/53194.html

Both mention the oddity of Microsoft Excel.

 

[greenLED]

Both mention the oddity of Microsoft Excel.

Excel is (in)famous for being full of computational glitches (and don't ever try to get a straight statistical answer out of it; even something as basic as its random numbers generator is faulty).

When I was finishing my dissertation I ran into a situation where running some basic arithmatic "backwards" to check my own math gave me (slightly) different answers. The issue drove me nuts for days! IIRC, prof helped me sort that one out - something to do with Excel rounding very large numbers improperly.

 

[Wits' End]

So no real authoritative answer. Or at least not one, real authoritative answer

So for clarity it doesn't hurt to use ()'s, and read the textbook before answering questions.