A button to despair
Some scientific calculators have a button for subtraction and one to change the sign of a number (circled above). However, when I enter negative 1 and press the x2 button I get an unexpected result. Why is that?
By the way, an online calculator like the one at http://www.calculateforfree.com/ behaves the way I like.
December 16th, 2009 at 2:13 am
Ah, one I KNOW the answer to.
When my learners first learn this, it really screws them up, but it does reinforce the idea of order of operations.
-2^2 really means (to a scientific calculator) -1*2*2 = -4
While:
(-2)^2 means (-2)*(-2) = 4.
The online calculator you linked to, doesn’t have the parenthesis options, so in order to fix it, the programmer chose to ignore order of operations. I spend about 5 minutes every year teaching this.
December 16th, 2009 at 4:48 am
“I enter negative 1″. What do you mean by that? In your picture, you ring the button marked (-). My TI-68 uses the symbol (-) to denote change-sign or unary minus, so I assume that Casio does likewise. (Time was when common practice on calculators was to use the symbol +/-.) Now, if I enter -1 on my TI-68, then square it, it gives me 1. On the calculateforfree calculator you linked to, there is a +/- key which is to be used as a postfix unary operator. Thus press in this order 1 +/- sqr. That gives 1, too. So that’s the way you like it. It’s the way I like it, too. So the Casio gives -1, I take it.
I daredn’t say the Casio is wrong; it’s just using a different philosophy, that’s all. On the calculateforfree, you change sign and square using postfix unary operators, so to square a negative number which you enter by sign and magnitude, you must apply first the postfix negate-operator, then the postfix square-operator, so the calculator couldn’t but give you (-1)²=1. The TI-68 has a different philosophy: of displaying the whole equation on screen for you, operators and all, before you press = to evaluate it. This compels it to honour conventions of mathematical notation by implementing many unary functions as prefix operators, and, because we put the minus sign to the left of the expression being negated, negation using the (-) key is one of them. We notate a power by a superscript to the right of the radix, so the TI-68 implements squaring by a postfix operator. So the designers could I suppose choose which of these operators has precedence over the other. TI-68 gave negation precedence over power, and the Casio, I presume, did the reverse. The Casio, I admit, follows mathematical notation better, because -x² means -(x²), not (-x)².
BTW the TI-68 honours the precedence of unary operators over binary ones, and the relative precedence of binary operators, OK. Two traps for the unwary:
2 + 6 / 2 * 3
Those who are used to calculators which don’t honour the usual conventions of precedence of binary operators will say 12. Those who remember that x/yz means x divided by the product of y and z will say 3. Correct is 11. (/ and * have equal precedence and associate from left to right. Multiplication by concatenation has precedence over division but this is a different operator from the calculator’s * operator.)
So you entered -1 and squared it, and got 1, I’ll be bound. That’s what my TI-68 does.
December 16th, 2009 at 10:04 am
I just checked on a TI-84 Plus Silver Edition. It gives negative one squared as -1, i.e. if you use these four keys: (-) 1 x^2 Enter, the answer is -1.
That TI-68 gives 1 is encouraging and odd. It uses integrated circuits Toshiba T9948A.
“I daredn’t say the Casio is wrong; it’s just using a different philosophy, that’s all.” I look forward to the day when my students use that line of reasoning when I hand them back their tests.