Oh sure, operationally a *real* negative number times any other *real* negative number is ALWAYS a positive number. But, if you draw a square with two numbers on a side of equal absolute value, one positive and the other negative, you will get a perfectly square graphic. It will appear in the lower left innnermost corner abutting the Cartesian Origin, yet the Math PhDs (and your grade school teacher) will tell you this figure is Not a square!
I once read that "minus numbers simply are not square" -- so I guess the *shape* of the analytic-geometry figure counts for nothing! Why not?
2006-08-15 18:11:51 · 13 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
Good question and the answer is negative numbers can be perfect squares. But in your approach considering absolute values should be given a rest, because once you consider an absolute value of a number you are no longer considering that same number, instead it's a mapping by the modulus function.
But you have an excellent point..If we're to represent complex numbers (numbers in the form a+ib, where i^2=-1) geometrically it would make sense to represent them on a plane rather than a line(we represent real numbers on a line), so one can conclude that real numbers are one-dimensional whereas complex numbers are two dimensional. Hence the anomaly in the Cartesian plane is resolved. Normal convention is to have multiples of i with real coefficients on the y-axis and real numbers on the x-axis.
But WAIT, didn't I just say i^2 = -1, there you are a minus number CAN be a perfect square! However I do not recommend underestimating the power of i, just because it's called 'imaginary'
, it's as real as it gets but to the 19th century one dimensional intellects imagining 2D was like us imagining 4D(or string theory), so unfortunately the name stuck.
eg. 2i * 2i = -4
3i * 3i= -9, etc.
2006-08-15 22:24:40 · answer #1 · answered by yasiru89 6 · 0⤊ 0⤋
yes, a 2 by 2 rectangular shape is a "square" no matter where it is... But you cannot draw a -2 by 2 shape! Ever tried going -2 meters? Impossible. There is no such thing as a negative distance. Distances are always postive. Mow I could interperet- 2 meters as going backwards 2 meters, but then I've introduced the concept of a direction to my distance. I still am moving a positive distance of 2 meters, but I just changed my direction. However this is more of an abuse of terminology. Your -2 by 2 square is kinda a figment of your imagination. You could again, have -4 acres, but again this requires the concept of debting or owing some postive amount. Or you could say you owe 2 areas of two acres... same thing.
You may want to ask instead, why is a negative number times a negative number a positive number? Other than, "things really work well that way and theres no (known) inconsistancy in handling numeric multiplication this way", I can't think of a good answer.
2006-08-16 01:43:38 · answer #2 · answered by Jay 3 · 0⤊ 0⤋
Whoa, guys, sure you can draw a square with negative sides. Try connecting the points [(0,0), (-4,0), (-4,-4), (0,-4), (0,0)] on Cartesian coordinates. The numbers in the ( ) are the (x,y) coordinates of each point in the figure. I put the (0,0) point in twice just to indicate, as you connect the dots, you come back to (0,0) to close the figure.
The illustrated figure is a square in the lower left hand quadrant with sides four units long each. Its area is 16 (= 4 X 4), the minus signs are dropped because we are only interested in length, not the direction of the sides.
The minus signs simply tell us the direction of those sides relative to the y or x axes. Minus x values mean left of the y axis for example. Plus y means above the x axis.
On the other hand, squaring any number, which is different from drawing a square, will give a positive answer unless delving into imaginary numbers (i). Despite their name, imaginary numbers have real applications in the real world of electronics, physics, etc.
2006-08-16 02:16:45 · answer #3 · answered by oldprof 7 · 0⤊ 0⤋
I think you interpretted what they were saying wrong. A negative number cannot be a square. meaning, no negative number can be the square of another number. No matter what number you square, positive or negative, the result is always positive. The only way to get a negative square is to use imaginiary numbers such as i ; (4i)^2 is -16, only because i^2=-1.
2006-08-16 01:21:19 · answer #4 · answered by John L 1 · 0⤊ 0⤋
The cartesian co-ordinate system does not measure length. It measures relative position from a point.
i.e., (-4,0) means 4 LEFT from the origin.
IT IS NOT A MEASURE OF LENGTH OR ANYTHING OTHER THAN POSITION
All lengths are positive.
No one ever said minus numbers were not square either. You are making stuff up or are simply confused. Dont act as if you proved anything or anyone wrong. You are what we call a nickel philosipher. Or perhaps a nickel mathematician. Because thats all your words and thoughts are worth.
EDIT
Walter they are called imaginary for a reason... and i think a mathematican would know better than an engineer.
2006-08-16 01:37:34 · answer #5 · answered by Anonymous · 0⤊ 0⤋
Hmmm, it has been a long time since I thought about such things, So you will have to research my answer..
You mentioned "absolute" numbers... are they not always positive, as in there are no negative absolute numbers?
I am thinking your "negative" square falls in the category of dividing by Zero or raising any quantity to a Zero power. It is a rule, provable mathematically. But the proofs have been proven so many times for so long it is just accepted as fact.
So I bet if you were to do a bit more research you would find the error is in your question, not in the proofs leading up to the rule.
Too late for me to take it any further. Good Question....
2006-08-16 01:28:53 · answer #6 · answered by skating265 2 · 0⤊ 0⤋
The square root of minus 4 is 2i, where i stands for the square root of minus one, an 'imaginary' number. What does this have to do with a Cartesian grid?
2006-08-16 01:18:22 · answer #7 · answered by Frank N 7 · 0⤊ 0⤋
Actually negative numbers have perfectly good square roots. We simply call these numbers "imaginary". But it really is all just a matter of definition; "imiginary" numbers are as "real" as any other number, as any graduate Electrical Engineer can attest to.
2006-08-16 02:06:23 · answer #8 · answered by walter_b_marvin 5 · 0⤊ 0⤋
Can you get a negative number from multiplying a negative by a negative or a positive by a positive? No.
2006-08-16 01:15:36 · answer #9 · answered by Stranger in a Strangeland 5 · 0⤊ 0⤋
the shape is a square but that is only measuring the absolute value of the area!
2006-08-16 01:15:43 · answer #10 · answered by Anonymous · 0⤊ 0⤋