Is i any less real than -1?

I don't think it is possible to actually comprehend i, since it is the square root of -1. We don't know of any numbers whose square equals a negative number, which is why i is imaginary. When you square a negative number, the negative signs cancel out, giving a positive value. In order to get a negative square, you would have to have some sort of crazy negative/positive amalgam or some type of half-positive-half-negative number.

imaginary numbers are ones that aren't possible in our normal, continuous mathematics. They are virtually always represented with i, since i is the basis for imaginary numbers.

I would say that yes, you can have negative somethings, even though you have to imagine them (which is why "imaginary numbers" might be a misnomer). For example, if you have six apples but you are hungry, you might think of your hunger as -2 apples.

have u heard of the number line?

well if u haven't then the number line is a 1 dimensional number system.

basically it's a straight line (say X axis) that contains all possible real numbers from -infinity to +infinity. any number located on this line is what we call a real number (even 1/2 lies on this line, if u have used a graph, it will be more clear)

the imaginary number 'i' = sqrt(-1) introduces another line (say Y axis) perpendicular to the first line we considered, which intersects the number line (X axis) at 0 (zero). the imaginary number thus introduces a 2 dimensional number system (a plane, say X-Y plane)

the value i = sqrt(-1) was encountered in mathematics and electrical engineering but it could not be plotted on the 1 dimensional number line. so we went in for the 2nd dimension that includes all multiples of 'i' along a line. the result was a plane in which every number ever encountered in mathematics could be plotted on a graph sheet

P.S.

sqrt(-1) = square root of -1

Your example of 1/2 indicates that you are missing the

difference between whole numbers and rational numbers.

Rational numbers can be expressed as the ratio of two

whole numbers, such as 1/2, 2/3 etc. There is a hierarchy

of numbers. The most restrictive are positive whole numbers.

More inclusive are positive and negative whole numbers.

More inclusive are rational numbers which include whole

numbers. So whole numbers are rational, but rational

numbers are not necessarily whole. On another separate

branch of numbers are the irrationals such as pi and e.

They cannot be expressed as a ratio of two whole numbers

and as a decimal, and they do not repeat. But irrationals

can still be compared to rationals(i.e., pi<4). The

inclusive set of rationals and irrationals are called real

numbers. The terms real and rational and irrational are

not meant to be interpreted philosophically or with every

day semantics, rather they're meant as a loose intuitive

label. So when we come up with the imaginary numbers let's

remember that imaginary is just a label. Now if we think

of the reals in one dimension(left and right on the number

line), then we can think of the imaginaries in another

dimension(up and down). So you can't compare imaginaries

to reals by the use of "greater than" or "less than". But

you can compare their relative magnitudes. For example,

2i has a magnitude of 2 and 3 has a magnitude of 3. So

3 is greater in magnitude than 2i.

The combined set of reals and imaginaries are called

complex numbers,(i.e., 3+4i) which, by the way has a

magnitude of 5. And instead of being on a number line,

they lie in the complex plane. And they have a "real"

purpose in math. For example, -1 has 3 cube roots as

you would of course expect. They are -1, 1/2+i(sqrt(3))/2

and 1/2-i(sqrt(3))/2 . Multiply them out, just for fun.

Now, finally getting back to English semantics,

negative numbers are not pessimistic, irrational numbers

are no less sane than rationals and imaginary numbers

do not dwell in the world of elves, fairies and unicorns.

Negative Numbers cannot have square roots because anything squared is ALWAYS positive NEVER negative. So the Imaginary Number System deals with(-1)^0.5 or the square root of One. IT is non-existant and is only a concept to work with!

It is easier to explain it with a number line. In theory, you can find and spot any real number on a number line, for instance, Pi, e, 3, 2.3443, 1/2, 1/3, and sqr(2), sqr(5). However, can you identify sqr(-1)on the number line? is between which two numbers? we call i imaginary number, because we dont really see it on a number line, like an imaginary friend.

Numbers are what they are they are, numbers. The "real" are very tangible in the sense they can represent money, speed, and weight just to name a few. Negative numbers are also used to represent things like debt. Just as the imaginary numbers are used to represent things in electrical circuits.

But when it comes to what is real and what is imaginary, simply put have you ever tripped over the number "1" or "i".

New types of numbers were needed over the history of mathematics for different situations. Originally there were the counting numbers and zero.

Then we needed to be able to solve for x + 1 = 0. Thats when negative counting numbers came about.

Then we needed to solve for 3x = 2. That is when the rationals arrived.

Then we needed to solve for x^2 = 2. That is when the irrationals came.

Then we needed to solve for x^2 = -2. Thus the complex numbers.

So new numbers arose when they were needed. There was a time when irrationals were incomprehesible to people, just like maybe the complex numbers are today.

These names are not intended to be a "definition" of real, negative, irrational, trascendent, imaginary... They are only "metaphoric". For example, "negative numbers" simulate a "negative income", i.e. a sum you are spending, when you are considering positives the sum you are receiving, and so on.

There are both imaginary. Unless you're an electrical engineer. Then i is real and j is imaginary.

they are very real in an electrical circuit containing inductnce/capacitance.

the axis is labled j by electricians to avoid confusion with current which is normally denoted by i. it is still sq root -1 though