Why is the square root of a decimal greater than the decimal value?
ex: sqrt .5 = .71 (rounded, naturally)
I asked my math teacher and he responded with the infinitely frustrating answer of 'it just is." Is there a more acceptable answer for an obsessive-compulsive level of curiosity, or is that really just how it is?
Because numbers between zero and one get smaller when you multiply them.
1/3 X 1/3 = 1/9 .33333 X .33333 = .11111
1/2 X 1/2 = 1/4 .5 X .5 = .25
1/5 X 1/5 = 1/25 .2 x .2 = .04
.71 X .71 = .5
A third of a third is a ninth
a half of a half is a quarter
a fifth of a fifth is a twenty fifth
It’s no different then when you learned fractions in grade school, you just never gave it any thought until now.
Well, it just is.
It might be easier thinking about it coming from the opposite direction. Multiplying one-half by one-half means dividing one-half by two, which results in one-quarter. In decimal terms it would be 0.5*0.5 = 0.25. Taking the square root is just the opposite.
Yeah, I’m seeing the logic there. I’ve been doing physics homework all night, and that is what stumped me. Despite it being entirely irrelevant to my actual homework. Thanks for the brain-prod.
PS: +5 pts for the ‘deal with it’ shades.
Think of squares and rectangles.
Think of a rectangle that’s 1" x 0.71". What’s the area of that rectangle? .71 square inches, right?
Then doesn’t it figure that if you shorten the long side to 0.71" to make it a square, that the rectangle would shrink? I mean, you’re shortening a side, after all. So now what’s the area of the .71" x .71" square? Surely, it’s less than our previous answer of .71. Turns out, it’s roughly .5. That makes sense that it’s a little smaller, since we only cut off a small chunk of the long line.
So apprently, .71 squared is .5. That means the square root of .5 is .71. Got it now?
Frankly I’m surprised your math teacher explained it the way that he did. Unless he was just tired or frustrated, there’s really no excuse for not explaining what’s really a pretty simple mathematical concept to a curious student.
He really is a great professor so I give him the benefit of the doubt. Most likely it had to do with the fact that my question wasn’t related to the topic at hand. In any case SDMB delivers. Thanks for the help.
I implied it above, but I want to be more explicit here: You should be wary of using the word “greater” if you’re just trying to understand this concept. Because if you attach units to those numbers, making them something tangible that you can actually imagine, you have to use different units. The .5 would have to have a square unit, like square feet or square miles. It’s two-dimensional. The .71 would have a linear unit like feet or miles.
So it makes no sense to say that .5 square miles is less than .71 miles. Those units are incomparable. It’s like asking which is bigger, an acre or a marathon, or a liter of milk vs. its boiling point… it makes no sense.
This. Multiplying by a fraction isn’t really multiplying, it’s dividing.*
As an exercise in algebra, try solving the inequality x² < x.
[SPOILER]x² < x
when x² - x < 0
when x(x - 1) < 0.
For x(x - 1) to be < 0 (i.e. negative), the two factors x and x-1 have to have different signs.
Either x < 0 and x - 1 > 0 (which is impossible)
or x > 0 and x - 1 < 0 (which implies x < 1).
So x² < x precisely when x is a number between 0 and 1.[/SPOILER]
Right, but it also makes no sense to say “a number between 0 and 1” (what the OP calls “a decimal”) except when dealing with “dimensionless” numbers.
When you add and subtract things, zero is special. It has no effect. If you keep dividing a number by a big divisor, the result keeps getting closer to zero.
In multiplication and division, it is one that is special this way. If you keep taking a number’s root, the result keeps getting closer to one.
Why the face? What counts as a height between 0 and 1?
Chessic Sense is not being pedantic. (well, at least not only!) It will serve you well to work at keeping your units straight and working the units in parallel as you work the numbers. It is a great way to catch errors, and can often suggest how to solve the problem if you can’t work it out otherwise. I recall this saving my grade in freshman physics, as there was a problem* on the final that had me stumped. I knew the answer had to come out as seconds though, so I just multiplied and divided by the available information until I ended up with everything but seconds canceling, and damned if it didn’t turn out to be the right answer. I don’t recommend this, but it was my only option at the time.
*Almost 30 years later I still remember the problem:
The Chordal Rapid Access Pineapple Company (CRAPCO) has bored a straight hole from Honolulu to New York. They have lined the hole with a frictionless material, and are able to maintain it at perfect vacuum. Pineapples are dropped into the chordal hole by the Hawaiian CRAP dropper, and caught by a CRAP catcher in Manhattan.
If the catcher in Manhattan is taking a CRAP break, and fails to catch the pineapple, it will fall back to Honolulu, and then back to Manhattan, and so forth. What is the period of this oscillation?
Insert dimensioned cartoon drawing of earth, with leigh and grass skirt wearing woman at one end and Yankees jersey wearing man at other. Dave Olson had started out as an Art major before deciding he liked Physics. He made what could have been a very dry subject fun.
Amen, Kevbo! Whenever my students ask me what the units are on the answer to some calculation, I always tell them “If you never take the units out of your calculation in the first place, then you never have to put them back in.”
And your method of “solving” a problem by dimensional analysis is used a lot more than you’d think, and by professional physicists. You’ve heard of the Planck units, for instance? There’s no actual physics that goes into deriving those, just dimensional analysis. The technique works so well that we trust it, more or less, even when we have no clue as to the actual answers.
Why couldn’t there be heights/lengths between 0 and 1? Like in Chessic Sense’s example, 0.71 feet or 0.71 miles
0.71 feet is between 0 and 1 foot. 0.71 miles is between 0 and 1 mile. But 1 foot is not the same as 1, and neither is 1 mile (and, of course, 1 foot is not the same as 1 mile).
Length (on one natural ideailzation) is a different one-dimensional space than the space of real numbers, with no distinguished element 1. You can multiply a length by a real number to get another length (as in multiplying a foot by 0.71 to get 0.71 feet), but lengths themselves can’t be compared to real numbers, as such.
Okay, I see what you’re getting at now.
Yeah, I see your point, but the reason I’m attaching units is not to preach the importance of doing the unit math, but to give a strategy for understanding simple yet confusing topics. I could just have easily said “Wouldn’t you agree that 71% of .71 must necessarily be less than 100% of .71?” It’s not so much about the fact that in this specific case, the units are different, as it is that it’s easier to think about if you talk the area of a square that you can see, draw, or walk around it. It’s easier if you think “Oh, yeah, duh…if I’ve got X change in my pocket, then such-and-such must obviously result from Y events.”
I don’t think they preach that enough in school. Keeping terms in the abstract is an easy way to ensure that students will data-dump it after the test. If you teach them to think about, say, derivatives as rates of change of car speed or height or growth, they’ll always remember what a derivative is. If you teach the distributive property as, say, piles of coins that get multiplied collectively, they’ll have no problem doing a sanity check later on when they think “Wait, is the rule that the 4 gets carried over every term, or just one and it disappears?” They might still forget the specific equations, but at least they’ll know what wiki article to conjure up.
I’ll get off my soapbox now.