which is greater? square root of negative 2 or square root of negative 4?

Answer 1

The simple answer is that the square root of negative 4 gives a larger imaginary number.
The square root of a negative number gives you a complex number, consisting of a real and imaginary part. The real part tends to come into play when you have vectors involved. The imaginary part is represented by i, where i is the square root of -1. Therefore, the square root of -2 gives 1.41i, while the square root of -2 gives 2i.

Answer 2

Technically, negative numbers do not have a square root since any number squared produces a positive outcome. A positive times a positive and a negative times a negative yield a positive. The way in which to express these numbers are what's called imaginary numbers, where the letter i represents the square root of negative 1. therefore the square root of negative 2, or -2^(1/2), is equal to (2i)^(1/2) and the square root of negative 4, or -4^(1/2), would be 2i. The magnitude of 2i is greater than [(2)^(1/2)]i, and since these represent negative numbers, that would make the square root of negative 2 the larger number since it is less negative.

The statement that says that these numbers can not be compared is a false statement, and here is why.
The eulers formula, which says e^(ix)=cos(x)+i(sin(x)). It is true that i is not a real number and can not be placed on a number line, but we know from this formula that there is a number x and a number i that exists. since we have 2i and [2^(1/2)]i, we can use the inequality that 2>2^(1/2) by instituting simple algebra dividing the inequality 2i>[2^(1/2)]i and dividing both sides by i.

This logic works. People assume that since we are taking a square root of a negative number that we will get a negative answer. this is a false assumption. the number i has only one representation, and that is graphically. The number i is not a real number, therefore it is not positive or negative, however comparable to other imaginary numbers. the graphical representation of these numbers would be that 2i=(2,0) and [2^(1/2)]i=(2^(1/2),0). Therefore we can finally conclude that the square root of -4 is greater than the square root of -2.

Imaginary numbers are quite a fascinating subject. eulers formula actually forms the unit circle and both the trigonometric functions sin and cosine can be represented with imaginary numbers. They are used in computer programming, electrical engineering, and in subjects such as quantum physics and electromagnetism. Imaginary numbers also have applications to geometry. The term imaginary is sort of a misleading term since they are applicable to real life problems. Numbers like -2 or 1/3 have no application to counting objects, but are abstractions themselves just like imaginary numbers. When you realize that all numbers themselves are abstractions than you can understand the logic that explains why this answer turns out to be the way it is. Hope that this fully explains everything.

Answer 3

-16/2, -4, 1, 19/5, square root of 49 On a number line, the numbers furthest left are the least. So, the negative numbers would be first. -16/2 is the same as -8, which is less than -4. 1 would be next because it's greater than -4, but less than 19/5 (which is 3 and 4/5) and finally the square root of 49 (7). and, same idea on the other one... -47, -13, 19/6, square root of 144

Answer 4

Both are imaginary numbers, because you cannot take the square root of a negative number. The square root of negative 2 is approximately 1.4i and the square root of negative 4 is 2i. If that's the case, then 4i should be the greater of the two.

Answer 5

Actually none of the current answers is completely correct.

About 1/2 the people got 1/2 the answer. You can take the square root of a negative number. These are called imaginary numbers (despite the name of the math concept, imaginary numbers actually do have meaning and are used very frequently in real science used to build today's technology). That is perfectly acceptable.
sqrt(-2) = i * sqrt(2) = i * 1.4142...
sqrt(-4) = i * sqrt(4) = i * 2

That part is what about 1/2 the people got correct.

The part no one answered completely correctly is it is also perfectly acceptable to compare imaginary numbers. Numbers are all complex numbers. A complex number has a real part (the part we are most familiar with day to day) and an imaginary part (the part being discussed here). So a complex number can be represented as "a + bi" where "a" is the real component and "b" is the imaginary component.

To display this think of a chart. The real component is your x-axis (horizontal) and the imaginary part is your y-axis (vertical). All numbers could be placed on this chart.

The sqrt(-2) and sqrt(-4) would be placed at (0, 1.4142) and (0, 2) respectively. These two points can be compared using vector algeabra. In simplest terms that would be an arrow drawn from the origin (0, 0) to the point in question. In this comparison what we would be measuring is the distance each of those points is from the origin. sqrt(-2) is +1.4142... units from the origin and sqrt(-4) is +2 units from the origin. The sqrt(-4) being further from the origin in the positive direction (to the right or to the top in our chart) makes it greater than a number farther to the left or bottom in our chart.

Answer 6

Both are imaginary numbers. I don't know if you can say that 2i (the square root of -4) is greater than 1.414i (approximately the square root of -2), because i is the square root of -1 and is neither positive or negative.

Answer 7

well actually, you can take the square root of negative numbers but you have to use the imaginary unit i which is defined as the square root of -1 and is a complex number.

So the square root of -2 is square root of 2 times i.

And respectively the square root of negative four is 2i.

To your actual question:
2i is smaller then square root of 2 times i (because i being the square root of a negative number. So the square root of negative four is smaller than the square root of negative 2

Answer 8

The answer would be the square root of -4.

PROOF

A square root is a number that when multiplied by itself is equal to the root number, (the number in question).


Sorry I don't have the symbol available so I'll use the symbol SR instead.

Example the SR of 4 is 2 or -2, 2x2=4 or -2x-2=4. In a negative square root problem there are no numbers that when multiplied by themselves eqaul a negative number. To get -4 you have to multiply a negative number by a positive number -2x2=-4, whereas 2x2=4or -2x-2=4. Since the only way to get that negative square root,(root number, SR-2 or SR-4 in this problem) is multiplying a negative by a positive they are not the square root, because they are not the same number multiplied by itself.

The answer comes from the use of imaginary numbers. The symbol for an imaginary number is i. i=SR of -1.
The SR of -4=2i, the SR of -2=1.41i.

You prove this by multiplying the square root 2 by itself 2x2=4, then multipling it by i,(square root of -1) giving you square root of -4. 2x2=4 4 x i=sr of -4

So back to the answer you would have 1.41i and 2i as answers. Since you are multiplying i by the coefficient in both cases the i's can be canceled out and not affect the absolute value. 2 >1.41, hence SR-4 > SR-2

On imaginary numbers, they have many uses in science, advanced mathematics, (it is a fundamental therom of algebra, as real numbers do not answer all polynomial equations) and especially in electrical engineering. They are not just an obscure irrelavant concept or a, "tricky" way to solve negative SR problems. Google them, you'll read all day if you don't get bored first.

ADD ON
imaginary numbers are complex numbers. The wiki for complex numbers gives a great dissertation.

2nd ADD ON
Patrick and others brought up good points on the meaning of greater. If you wanted to know the absolute value, (distance from 0 on a numberline), then the SR -2 with a absolute value of 1.41 would be larger than the SR -4 which would have an absolute value of 2.

If you wanted to know which was greater in solved terms, SR-4 is >SR-2.

SR-4=2i SR -2=1.41i remove the i or imaginary part of the answer by multiplying each by -i which in reality is -SR-1. If you do that you are left only with the real number part of the complex number. In that case 2>1.41, hence SR-4 is greater.

Answer 9

The square root of negative numbers actually gives you what they call "imaginary" numbers.. So the square root of -2 is actually 2i, just as -4 is 4i. You can't actually TAKE a square root of a negative number and have it be real.

So the answer is: neither.

Answer 10

The square root of negative 2.