# [General]km-2

Status
Not open for further replies.

#### 4ania4

##### Junior Member
Dear Teachers,

How should I read this :

3 million tonnes km[SUP]-2 ?

[/SUP]
[SUP]Is this the same as the following:
[/SUP]

3 million tonnes/km[SUP]2 ?

4ania4[/SUP]

#### emsr2d2

##### Moderator
Staff member
Dear Teachers,

How should I read this :

3 million tonnes km[SUP]-2 ?

[/SUP]
[SUP]Is this the same as the following:
[/SUP]

3 million tonnes/km[SUP]2 ?

4ania4[/SUP]

After "3 million tonnes", does it really say "km-2"? I'm not sure if you meant for the letters and numbers in the two versions to be identical, with only the addition of the "slash" in the second. If so, the first is rather unnatural. I would expect to see the slash.

Three million tonnes per square kilometre.

#### 4ania4

##### Junior Member
After "3 million tonnes", does it really say "km-2"? I'm not sure if you meant for the letters and numbers in the two versions to be identical, with only the addition of the "slash" in the second. If so, the first is rather unnatural. I would expect to see the slash.

Three million tonnes per square kilometre.

Thank you. It must be really rare but I have found a few examples of it in a text on tropical cyclones in Cambridge Academic English Upper Intermediate,

e.g. In total the passage along a coast can induce a change in load on the Earth's crust of 10 million tonnes km[SUP]-2[/SUP].

Last edited:

#### emsr2d2

##### Moderator
Staff member
I have no idea what a "minus square kilometre" is then, if indeed the "dash" is intended to be a minus sign. I think we need a scientist or a maths graduate!

#### SoothingDave

##### VIP Member
Present!

Note that the negative 2 in the posted example is in superscript. That is, it is an exponent of negative 2, not a subtraction of 2.

Negative exponents indicate that the value belongs on the bottom of a fraction.

So it does mean "per square kilometer."

It's quite unusual in my experience to see negative exponents used like this. Perhaps it was a typesetting decision, but it's strange.

In my field we see "kg/cm^2" for kilograms per square centimeter. (or "kg/cm2" where it is understood that the "2" is an exponent.)

Last edited:

#### 4ania4

##### Junior Member
Thank you so much. Now the meaning is clear to me. I think that if you were to read "km[SUP]-2" [/SUP]aloud, you would just say "per square kilometres". Am I right?

Last edited:

#### emsr2d2

##### Moderator
Staff member
Present!

Note that the negative 2 in the posted example is in superscript. That is, it is an exponent of negative 2, not a subtraction of 2.

Negative exponents indicate that the value belongs on the bottom of a fraction.

So it does mean "per square kilometers."

It's quite unusual in my experience to see negative exponents used like this. Perhaps it was a typesetting decision, but it's strange.

In my field we see "kg/cm^2" for kilograms per square centimeter. (or "kg/cm2" where it is understood that the "2" is an exponent.)

So does "kg/cm^2" mean "the number two should be in superscript but we couldn't do it for some reason"? I have not the first idea what "an exponent of negative 2" means, by the way!

#### Rover_KE

##### Moderator
Staff member
'...per square kilometre'.

#### emsr2d2

##### Moderator
Staff member
Thank you so much. Now the meaning is clear to me. I think that if you were to read "km[SUP]-2" [/SUP]aloud, you would just say "per square kilometres". Am I right?

Singular - per square kilometre.

#### non_e_giusto

##### Member
This is mathematics, not English. So as a chemist I dare reply to this thread.

As stated above, the minus sign indeed means that it is part of the denominator of the fraction, and here is why:

Let's do some simple exponentiation:

2[SUP]1[/SUP] = 2
2[SUP]2[/SUP] = 4
2[SUP]3[/SUP] = 8
2[SUP]4[/SUP] = 16

You see that every time you add 1 to the exponent, the outcome doubles (exponentiation works like that). But you can also see it the other way around: every time you subtract one from the exponent, you divide by two. So we can do what we just did, but then in reversed order:

2[SUP]3[/SUP] = 8
2[SUP]2[/SUP] = 4
2[SUP]1[/SUP] = 2
but we don't have to stop here, we can simply keep dividing by 2, so:
2[SUP]0[/SUP] = 1
2[SUP]-1[/SUP] = 1/2
2[SUP]-2[/SUP] = 1/4
2[SUP]-3[/SUP] = 1/8

So what do we see? If you have a negative exponent, it's basically 1 divided by the outcome of the positive exponent (2[SUP]1[/SUP]= 2 and 2[SUP]-1[/SUP]= 1/2).

1/2 (2[SUP]-1[/SUP]) is called the mathematical inverse of 2. The reason is that dividing by 2, is the same thing as multiplying by 1/2 (and vice versa). It also works with symbols: multiplying with s[SUP]-1[/SUP] , is the same thing as dividing by s. Scientists use this a lot, because it allows you to neatly write entire fractions in just one line. Instead of meter per second m/s, they write: m s[SUP]-1[/SUP] = m * 1/s = m/s

I hope this is clear, otherwise, ask away.

Last edited:

#### SoothingDave

##### VIP Member
So does "kg/cm^2" mean "the number two should be in superscript but we couldn't do it for some reason"? I have not the first idea what "an exponent of negative 2" means, by the way!

Yes. That is how programmers and engineers and such express exponentiation, because we can't type in a superscript into programming languages or when typing simple text messages.

#### SoothingDave

##### VIP Member
You see that every time you add 1 to the exponent, the outcome doubles (exponentiation works like that).

It doubles because you are exponentiating two, not as a general rule.

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81

This tendency for values to rise rather quickly is why very rapid growth is called "exponential" in common use of the language.

#### non_e_giusto

##### Member

It doubles because you are exponentiating two, not as a general rule.
True. In my example it doubled because I was with a base of 2. If you have a base of 3, it triples. The general rule is that adding 1 to the exponent means multiplying the outcome with the base number. I was intentionally keeping it simple for didactic purposes, but you are totally correct.

Status
Not open for further replies.