Mathematical notation question-JR High

[Marduke]

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.

It's an international standard agreed upon by mathematicians around the world. How much more authority do you need??

 

[greenLED]

It's an international standard agreed upon by mathematicians around the world. How much more authority do you need??

Which, given the wide range of answers given here, seems like it hasn't trickled down to us mere mortals?

Or, some of us are too old to remember basic algebra.

 

[effulgentOne]

As a math major, who had Jr-High math less than a decade ago poke, teachers/textbooks often do weird things, but:
-5^2 = -25
(-5)^2 = 25

For Example, consider the functions y = x^2, and y = -x^2. Here, x is always treated the same way as 5 is above. y = x^2 is always nonnegative, and y = -x^2 is always nonpositive, they are not equivalent.

y = x^2 = (-x)^2 != -x^2

 

[Empath]

That doesn't help, effulgentOne. It applies only when x is positive. If x is negative, such as -5, then it isn't applicable.

The claim of -5^2=-25 is in error, because -5 is not an operation. -5 is a value.

 

[Marduke]

The claim of -5^2=-25 is in error, because -5 is not an operation. -5 is a value.

Not according to international convention. See above references. -5 would only be a value if it was in parenthesis. Without them, it's part of a larger expression. In mathematical expressions, exponents are ALWAYS done before negation.


But I understand the confusion. If you verbally say "negative five squared", the answer is 25 because the grouping is implied in the statement. However, evaluating the written expression -5^2, "-5" is no longer a value, it's part of an expression -1*5. Written expressions follow very strict rules.

Edit:
Verbally, "negative 5" is a value. Written, -5 is an expression, subject to mathematical conventions.

Think about this:
evaluate the expression following order of operations: 0 - 5^2 = ?

 

[43X16]

Yes, absolutely. (Btw in a TI & the like you do get -25 for -5^2. This is the convention.)

Not according to international convention...
... Written expressions follow very strict rules.

Edit:
Verbally, "negative 5" is a value. Written, -5 is an expression, subject to mathematical conventions.

Think about this:
evaluate the expression following order of operations: 0 - 5^2 = ?

 

[asdalton]

Yes, absolutely. (Btw in a TI & the like you do get -25 for -5^2. This is the convention.)

My TI-89 does the same thing, even though it uses a separate key for the unary negation operator.

The TI-89 even reprints the entered expression -5^2 as -5², and returns -25 as the answer. (But be warned that if you are using a simple calculator that displays only one number at a time, and you press the <x²> key when -5 is displayed, the calculator will treat -5 as a unit and perform (-5)² = 25.)

There's a reason why MS Excel has been singled out as an oddity; other programming languages (including MS Visual Basic macros in Excel!) and mathematical convention do the exponentiation before negation.

 

[Marduke]

Think about this:
evaluate the expression following order of operations: 0 - 5^2 = ?

I'll give some examples which illustrate why it is so important to always follow the correct order of operations.

The following equations are identical, and equal -25

-5^2 = -(5^2) = -25
0-5^2 = 0-(5^2) = 0-(25) = -25
3-5^5-3 = 3-(5^2)-3 = 3-25-3 = -22-3 = -25

Now, what happens if you treat -5^2 as (-5)^2, as incorrectly described by some above? Let's do all these equations assuming negative numbers are values instead of expressions. "-5" is now a value, always assumed to be (-5)

-5^2 = (-5)^2 = 25 no surprise what happened there

0-5^2 = 0(-5)^2 = 0*25 = 0 wait a minute!! same equation, different answer??

3-5^2-3 = 3(-5)^2-3 = 3*25*(-3) = 75*(-3) = -225 a third different answer!?!?

As you can see, if you assume all negative numbers in equations to be "values" instead of "expressions", you suddenly loose all ability to subtract!! What type of world do we live in where subtraction suddenly doesn't exist?




I for one personally welcome our addition-only overlords...

 

[Empath]

Agreed. The expressions must be written in a way that dictates logical and absolute interpretation.
Considering that -x^2 =/ x^2, a negated expression would be understood unless written as (-x)^2.
Then it would be (-x)^2 = x2.

or

(-5)^2 = 25

For clarity though, the textbook expression should have been written as -(5^2) = -25

Now, I'm supposed to face the camera and say "my name is Empath, and I'm not smarter than a fifth grader".

 

[Wits' End]

Now, what happens if you treat -5^2 as (-5)^2, as incorrectly described by some above? Let's do all these equations assuming negative numbers are values instead of expressions. "-5" is now a value, always assumed to be (-5)

-5^2 = (-5)^2 = 25 no surprise what happened there

0-5^2 = 0(-5)^2 = 0*25 = 0 wait a minute!! same equation, different answer??

3-5^2-3 = 3(-5)^2-3 = 3*25*(-3) = 75*(-3) = -225 a third different answer!?!?

As you can see, if you assume all negative numbers in equations to be "values" instead of "expressions", you suddenly loose all ability to subtract!! What type of world do we live in where subtraction suddenly doesn't exist?

I for one personally welcome our addition-only overlords...

How about 0- -5^2 ? =25 = -25 or =0 ?

 

[asdalton]

How about 0- -5^2 ? =25 = -25 or =0 ?

This is unusual notation, but both the TI-89 and Visual Basic give 25, with Excel giving the expected pathological answer of -25. (Excel parses -5 as a single number, squares it, and then subtracts the result from 0.)

Either way, these computer programs treat the first "-" as a subtraction sign and the second as a unary negation operator.

By the way, expressions like 3*-5 are definitely allowed (answer: -15), consistent with negation being treated like multiplication by -1.

 

[Marduke]

How about 0- -5^2 ? =25 = -25 or =0 ?

That (non-standard) notation would, by strictly following order of operations, yield 0--(5^2) = 0-(-25) = 0+ 25 = 25

But if you created the original expression, good luck finding a math professor who would not mark it wrong for not using standard notation to begin with, and using parenthesis to make your intentions clear.

 

[greenLED]

"my name is Empath, and I'm not smarter than a fifth grader".

What Empath said!

 

[Wits' End]

Subject came up in our house again. I still don't recognize -25=-1*25 [as opposed to (-25)] all the time but will concede the point. Anyhow here are some fun, to those who like math at least, links.


http://acronyms.thefreedictionary.com/PEMDAS

PEMDAS Please Excuse My Dear Aunt Sally
PEMDAS Parenthesis, Exponents, Multiply, Divide, Add, Subtract
PEMDAS Pink Elephants Make Dandy Apple Sauce
PEMDAS People Eat More Donuts After School
PEMDAS Purple Eggplants Make Delicious Afternoon Snacks
PEMDAS Parenthesis Exponents Multiplication Division Addition and Subtraction
PEMDAS Peanuts Ended My Donut Addiction Saturday

http://mathstory.com/Poems/PEMDAS.html

[SIZE=+4]PEMDAS[/SIZE] [SIZE=+4]PEMDAS!!!
[/SIZE][SIZE=+2] Yelled teacher,
Voice loud like chimes,
Know what comes first,
Subtraction or times...
Or division, or adding, or any old math,
If you can't learn your [/SIZE][SIZE=+4]PEMDAS[/SIZE][SIZE=+2],
You can't pass my class!!!
The name of the teacher was Mary McGrath,
And getting math wrong,
Incurred all her wrath!
And [/SIZE][SIZE=+4]PEMDAS[/SIZE][SIZE=+2] she said,[/SIZE] [SIZE=+2]Makes you so right,
When you've seen [/SIZE][SIZE=+4]PEMDAS[/SIZE][SIZE=+2],
You've seen the light...[/SIZE]
[SIZE=+2]Let's learn it again,
The order of math,
[/SIZE][SIZE=+4]PEMDAS[/SIZE][SIZE=+2],[/SIZE][SIZE=+2] so fine,
Like pockets of cash!!!

[/SIZE][SIZE=+4]P[/SIZE][SIZE=+2]arenthesis first,
[/SIZE][SIZE=+2]Subtracting is last...
Multiplication, Division,
The vote has been cast...[/SIZE]
[SIZE=+4]E[/SIZE][SIZE=+2] for [/SIZE][SIZE=+2]exponents,
[/SIZE][SIZE=+2]You already know,
Let us say, [SIZE=+2][/SIZE]
Makes numbers grow...
[/SIZE][SIZE=+4]M,[/SIZE] [SIZE=+2]multiplies[/SIZE][SIZE=+2],
Before you go add,
[/SIZE][SIZE=+4]D's[/SIZE][SIZE=+2] for [/SIZE][SIZE=+2]division[/SIZE][SIZE=+2],
Not [/SIZE][SIZE=+2]Dave, [/SIZE][SIZE=+2]Dog[/SIZE][SIZE=+2] or [/SIZE][SIZE=+2]Dad[/SIZE][SIZE=+2]...
The [/SIZE][SIZE=+4]A's[/SIZE][SIZE=+2] for [/SIZE][SIZE=+2]addition,[/SIZE][SIZE=+2]
And addition's no fad,
[/SIZE][SIZE=+4]S[/SIZE][SIZE=+2] for [/SIZE][SIZE=+2]subtraction,[/SIZE][SIZE=+2]
Who's last but still rad...[/SIZE]

[SIZE=+2]Part II[/SIZE]
[SIZE=+2]That same afternoon,
In our math class,
Something bad happened,
With Mary McGrath...
An innocent boy,
Joey Zach Zemjas,
Discovered what happens,
When you forget [/SIZE][SIZE=+4]PEMDAS[/SIZE][SIZE=+2]...
Now I can say,
McGrath is real mean,
Resident evil,
Mean-teacher-queen!!![/SIZE]
[SIZE=+2]But back to my story,
And Joey's[/SIZE][SIZE=+2] big blunder,
At least he's alive,
And that's a true wonder...[/SIZE] [SIZE=+2]McGrath made a problem,
[/SIZE][SIZE=+2] For us, to be solved,
But Joe didn't care,
He wasn't involved...[/SIZE]
[SIZE=+2]T[/SIZE][SIZE=+2]he problem she'd written,
4 X 3 - 2...
Wasn't so hard,
If you just thought it through...[/SIZE] [SIZE=+2]But Joey had trouble,
He made a mistake,
His [/SIZE][SIZE=+4]PEM-DAS[/SIZE][SIZE=+2] was raw,
Like the batter of cake...[/SIZE]
[SIZE=+2]Joey said,
3-2 is 1,
And 4 X 1 is 4,
Four's the answer,
I'm satisfied,
I'm feeling pretty sure...

McGrath grabbed Joey's little hand,
And yanked him from his seat,
You ignored the rules of [/SIZE][SIZE=+4]PEMDAS[/SIZE][SIZE=+2],
Now it's time for your defeat!!!
[/SIZE][SIZE=+3]PEMDAS-SCHMEMDAS-GEMDAS-GOO![/SIZE][SIZE=+2]
Yelled Joey to the class,
McGrath turned red,
McGrath turned blue,
[/SIZE][SIZE=+2]This day will be your last!!![/SIZE] [SIZE=+2]T[/SIZE][SIZE=+2]wisted Joey's left arm,
Back behind his head,
Probably should have hurt a lot,
But Joey laughed instead...
Then she bent him silly,
Glued him to the wall,
Used a pound of super-glue,
So he'd never fall...
Parts of Joe she twisted,
Twisted him to spell,
[/SIZE][SIZE=+4]PEMDAS, PEMDAS, PEMDAS [/SIZE][SIZE=+2]
She felt so very swell...
Joey hung all last year,
'Til the last full-day of school,
And when she took him down from there,
Joey knew the rule...

I hope you all have listened,
I hope you've all tuned in,
I hope you know your,
[/SIZE][SIZE=+4]PEMDAS[/SIZE][SIZE=+2]
It's the only way to win...[/SIZE]

[SIZE=-1]P.S.[/SIZE] [SIZE=-1]4 X 3 - 2...[/SIZE] [SIZE=-1]is equal to 12 - 2 which is 10...[/SIZE]
[SIZE=-1]Multiplication comes before subtraction...[/SIZE]

 

[HarryN]

Funny - my engineering background (admittedly a while back) would have made that expression treat the -5 as a singular number. Perhaps more importantly, since engineering calculations can get quite long, I would have carefully placed ( ) to make sure it was not misinterpreted.

Thanks for bringing this point up - maybe it will help my kids out sometime.

My older daughter is a HS math teacher in training - just for fun, I will ask her. (not that we actually agree on that many things )

edit /add

So my daughter, the HS math teacher's reaction:
- Is there a parenthasis ? - answer - NO
- Only a fool would write an equation this way, as it just would confuse kids. - (we agree on something, but, ok- what is the answer ?
- The equation is treated as (-1) x (5) x (5) = -25

Thanks for starting this thread, I had a good reason to call her.

 

[Wits' End]

Please see my new question at the end of the first post