# Why is -10 smaller than -100?

My physics teacher told me recently that -10 is less than -100.

The way I look at it is in 2 ways:

1. Having minus £100 would mean you have less money that if you had -£10.
2. If you are presented with a number spectrum ranging from -100 to 100, it is clear that -100 is lowest down the spectrum and therefore lower/less that -10.

So is there any valid explanation as to why is -10 less than -100?

• Both your examples (and some of the answers below) show that -100 is less than -10. This is true, as you can easily check using the definition of less than. The terms less than and greater than are very standard; everyone should use them in the same way. I consider a statement like "-10 is smaller than -100" to be more ambiguous and informal. For example, if we set up a coordinate system and use it to describe forces of -10 and -100, then the force of -10 is the smaller force. You should use the context to determine if "smaller than" means "less than" or "smaller in magnitude." – Jonas Kibelbek Oct 9 '12 at 16:26
• Yes, I should really have mentioned that in this case it is smaller in magnitude, as he was talking about the current through the circuit. – ODP Oct 9 '12 at 16:29
• My friend just said, "I think -10 is less that -100 because it's closer to 0 and 0 is as low as possible" - any counter arguments? can't think of one. – ODP Oct 9 '12 at 16:33
• "Zero is not as low as possible" – Dan Neely Oct 9 '12 at 19:20
• Send your physics teacher here. – Preet Sangha Oct 10 '12 at 1:49

## 6 Answers

Your teacher is wrong or you misheard him/her. $x < y$ means $x$ is to the left of $y$ on the number line, when $x$ and $y$ are real numbers. Since $-100$ is to the left of $-10$, $-100 < -10$.

If you want to talk about magnitude, then you're comparing absolute values, i.e., the distance from $0$. In that case, we're comparing $|-100| = 100$ and $|-10| = 10$. And, $10 < 100$ so $|-10| < |-100|$. But, if that is what you are talking about, you would need to mention something about magnitude or distance from $0$.

• He was talking about magnitude as it was when doing an experiment involving the current of an electrical circuit - therefore he is correct – ODP Oct 9 '12 at 16:30
• @OllyPrice Yes, so he was correct and you heard him correctly but there was a slight misunderstanding as to what exactly he was referring to? – Graphth Oct 9 '12 at 17:45
• @Graphth I could of course be wrong, but I believe the OP is just nitpicking. He/she clearly understands magnitude and is in a class where they are taught about currents. They knew what the teacher was saying. I think. – im so confused Oct 9 '12 at 17:55
• ah sorry, wasn't clear – im so confused Oct 9 '12 at 17:56
• It's correct to say -10 is smaller or of lesser magnitude than -100, but "less" by itself doesn't mean smaller in magnitude. What words exactly did your teacher use? – Russell Borogove Oct 10 '12 at 0:12

I think your physics teacher is mistaken. -100 is unquestionably less than -10, because $-100 < -10$. If you owe \$100 you have less money than someone who owes only \$10.

However, -100 is bigger than -10, because if you owe \$100 you owe a bigger amount than someone who owes only \$10.

Addendum: Some commenters have claimed that "-100 is unquestionably less than -10, because $-100 < -10$" is tautological. It is not. The first clause concerns the meaning of the English-language phrase "less than", which is what the original question was asking about. The second clause concerns the formal mathematical statement that $-100 < -10$. I am asserting that the phrase "less than" is normally understood to mean the mathematical relation denoted by "$<$". The conventional reading of "$<$" as "less than" or "is less than" supports this claim. In contrast, in the following paragraph I am claiming that the meaning of the English-language phrase "is bigger than" (and implicitly, "is smaller than") is not modeled by the mathematical $<$ relation. (More precisely, it is modeled by $<$ only for non-negative quantities.)

These claims might be false, or poorly supported by evidence, but they are not tautological. I hope this clears things up.

• Huh? I've never in my life seen this usage of bigger before. Does bigger mean "larger as an absolute value" for you? Where is that definition taught? – fgp Oct 9 '12 at 16:19
• "-100 is unquestionably less than -10, because $-100 < -10$" This is a tautology.. – Mikko Korhonen Oct 9 '12 at 16:24
• Similarly: "one plus one is two, because $1 + 1 = 2$" – Mikko Korhonen Oct 9 '12 at 16:29
• “-100 is … less than -10, because $-100 < -10$” – what is this sentence supposed to mean? This is just “x because x”. – Konrad Rudolph Oct 9 '12 at 16:46
• Bigger is a vague term to which you can give a local definition in your discussion, lecture, paper or book. -100 is certainly more "magnitudinous" than -10, and so if you define "bigger" as "more magnitudinous" then it is bigger. – Kaz Oct 10 '12 at 4:55

It isn't, plain and simple. -100 is less than -10, i.e. -10 is larger than -100. Generally, $x$ is less than $y$ ($x < y$) if $x-y$ is negative.

Since $(-100) - (-10) = -100 + 10 = -90$ is negative, $-100$ is less than $-10$.

Your physics teacher was wrong, or he defined a non-standard less-than operation on the integers first. If you define your own meaning of less-than, you can obviously make it behave arbitrarily. Whether or not such a definition has any value is a different question, though.

• And if you apply $x$ to -10 and $y$ to -100? – ODP Oct 9 '12 at 16:13
• Then you're asking if $-10$ is less than $-100$. Since $(-10) - (-100) = -10 + 100 = 90$, the answer is no. Note that if you swap the values of $x$ and $y$ in $x-y$, the sign of the result changes. Thus you always have exactly one of $x < y$ or $y < x$, never neither and never both, since exactly one of the differences will have a negative sign. – fgp Oct 9 '12 at 16:17

By definition we say that a number $x$ is less than a number $y$ if the number $y - x$ is positive. So with $x = -100$ and $y= -10$ we have that $y - x = 90 > 0$, so by definition $-100$ is less than $-10$.

I am guessing that your teacher had different definition. You might want to ask a physics.stackexchange.com about whether such a definition would exist for some physics problem.

• It would seem that Graphth's answer is somewhat related to what my teacher was thinking about when he said it (absolute values). – ODP Oct 9 '12 at 16:12
• @OllyPrice: That might be what your teacher was talking about. So, yes I guess that you could define $x < y$ if and only if $\lvert x\lvert < \lvert y \lvert$. But then again, on the real line you would have $-10 = 10$.... which would be (IMO) a bit of a mathematical headache. – Thomas Oct 9 '12 at 16:15

There is an easier way to see this (not using inequality). We say that $x$ is less than $y$ if there is a positive number $z$ such that $y=x+z$ (in fact this is the definition of $x < y$ for real numbers).

In this case you can say $-100$ is less than $-10$ because there is a positive number $z$ such that $-10=-100+z$, of course $z = 90$.

As far as physics is concerned, it is a matter of context. Sometimes, negative numbers are just part of a scale, as I am sure that your teacher would agree that -100°C is less than -10°C and sometimes, negative numbers are a way of indicating directions, a negative current is actually just a positive current in another direction. The problem disappears if you leave the 1-dimensional world and note that the first kind of quantities are scalar and the second kind of quantities are vectors. One can employ a more precise language than your teacher, but there is no ambiguity in context. It is not useful to insist on unnecessarily formal terminology.