Is it true to say that all square roots have two numbers, both positive and negative?

Eg √36 = 6, because 6 x 6 = 36

But as negatives multiplied give a positive number is it also true to say that √36 also = -6, because -6x-6 =36?

Cheerz

GIAN🙂 XXX

(maths age about 12)

Not true. The square roots of positive numbers, yes. They are two-valued, with one value positive, and the other, negative.

For complex numbers, it's more involved. For negative numbers, you need complex numbers already.

4 hours ago, Gian said:

Is it true to say that all square roots have two numbers, both positive and negative?

Eg √36 = 6, because 6 x 6 = 36

But as negatives multiplied give a positive number is it also true to say that √36 also = -6, because -6x-6 =36?

Cheerz

GIAN🙂 XXX

(maths age about 12)

Because a -ve "real" number squared is +ve, as you point out, there is no way the square roots of -ve numbers can be "real" numbers. For example what is √-4 ? To deal with this, back in the c.17th, it was decided to invent a square root of -1 and call it i. Once you have that, you can say that √-4 has a value of ±2i. Numbers like this with i as a factor in them are called "imaginary" numbers, as opposed to 'real" numbers. Numbers can also have both real parts and imaginary parts, e.g 1 +2i. Such numbers are called "complex" numbers.

Graphically, you can plot the real numbers along a horizontal line, with zero in the middle, -ve numbers extending to the left and +ve numbers to the right, to form an 'x' axis. Imaginary numbers however would be be plotted at right angles, vertically, as if on a "y"axis, with +ve values upward and -ve values downward from the central zero. A complex number like 1+2i, with both real and imaginary parts, would be located on the graph 1 unit along the x axis to the right and 2 units up the y axis from the central zero or "origin". (Such a graph is sometimes called an Argand diagram, after the Frenchman who devised it.)

Although one might at first think the notion of defining i is perhaps a bit silly and pointless, the fact that you can plot complex numbers on a graph in this way turns out to make them enormously powerful as a tool in mathematics and science. You can represent one on the Argand diagram as a line drawn from the zero (origin) to the location I have described. This line has a direction and a magnitude. It is therefore a vector, with real and imaginary components at right angles to one another.

You can also define it in another way, by the length of the line and the angle that it makes with the x axis. If the length is r and the angle is 𝜽, the complex number can be written as r(cos𝜽 +i sin𝜽), where rcos𝜽 is the x component and rsin𝜽 is the y component. It turns out that angular representation opens the door to a host of applications in physics where there is "sinusoidal" periodic motion e.g. waves or the theory of alternating current in electrical circuits. There are also many connections in maths too, involving exponentials and so on but that's another story.

So that is why imaginary numbers are such a fruitful idea, even though they may at first glance seem crazy and just a mathematical curiosity.

(P.S. I write this as a chemist, not in any way a mathematician, so the way I express this may not be rigorous to a mathematician. It's just how I think of it. Actually, it was learning about complex numbers in the 6th form at school that first made me begin to enjoy maths. I just thought they were so cool in what they could do and how many areas of maths and physics they are able to link together.)