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In the mathematics,

a < b

I think it should be said as "a is less than b"

So, does can I say the title ("a < b < c") as

b is larger than a and less than c

or is there a better way to say?

rajah9
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curlywei
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  • If you are asking this with the intent of clear communication to some other person or groups of people, you could combine several answers. Say one answer, then "in other words," then another answer. For example. – Phlarx Feb 13 '20 at 17:33
  • Who is your audience? If you're talking to mathematicians, many of the answers below are ok if generally over-precise. If you're talking to non-mathematicians, many of the answers below are painfully inadequate. – Tony Ennis Feb 13 '20 at 20:48
  • What about : "a, b, c are sorted in increasing order"? – Eric Duminil Feb 13 '20 at 21:42
  • Related: I asked effectively the same question on SE Math Educators last month. To date there has been no consensus: https://matheducators.stackexchange.com/questions/17706/how-to-read-chained-equalities-out-loud – Daniel R. Collins Feb 14 '20 at 01:40
  • As a side note, I disagree that this is opinion-based. It's actually something I teach in my math classes with a first-day handout reasoning from basic grammar (which I wish I could share here, but currently can't with the question closed). Recommend this question be re-opened. – Daniel R. Collins Feb 14 '20 at 01:52
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    I don't think I understand how this is opinion based. There may be variation among subgroups but that's not a matter of opinion. – Mitch Feb 14 '20 at 02:48

9 Answers9

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I would say “a is less than b which is less than c”. Just saying “a is less than b is less than c” is ambiguous about whether it’s a or b that is less than c.

Mike Scott
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    “a is less than b is less than c” may or may not be ambiguous, but in the context of math, it is what is said. – jamesqf Feb 13 '20 at 17:28
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    I don't know how you could parse the second to mean anything other than a<b<c. Can you clarify why you think it's ambiguous? – Azor Ahai -him- Feb 13 '20 at 20:48
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    @AzorAhai I think it's more unclear than ambiguous, but once the expression is familiar it's no longer unclear either. – Daniel Feb 13 '20 at 21:31
  • @AzorAhai A valid interpretation of that phrase is "the value of a is less than value of c minus b". 'a is less than b; b is less than c' removes the ambiguity. – JimmyJames Feb 13 '20 at 22:06
  • “a is less than b is less than c” also sounds a little like it could mean "(a < b) < c", but I don't think < is generally defined for boolean values. – bdsl Feb 13 '20 at 22:25
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    @AzorAhai Consider the following (slightly longer sentence): "a is less than b, is less than c, and is greater than 0". That parses out as 3 simultaneous equations: a<b, a<c and a>0... It is then not entirely unreasonable to parse "a is less than b is less than c" (which is, regardless, grammatically incorrect) as a<b and a<c – Chronocidal Feb 17 '20 at 12:49
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    @Chronocidal I finally understood it, what helped me what parsing it as "a is less than (the difference between b and c)." So if b=10 and c=5, a<5. Thanks though – Azor Ahai -him- Feb 17 '20 at 18:22
12

In higher-level math, my experience says that a < b < c is usually pronounced as "a less than b less than c".

A large part of this is context. If we're examining the result of something, it's certainly possible that someone would say "b is between a and c", leaving some information out (that a < c). This is especially true where one or both of a and c are fixed, as in 2 < b < 7 ("b is between 2 and 7").

The most common case of a < b < c is when one is stating conditions, as "In the case a less than b less than c, we have...". It's easy to see why it's pronounced that way in this usage - we're naming the case we're referring to instead of talking about what the name of the case represents. Since we're just reading a name, we pronounce each character separately.

Note that the programming usage (the other place this might show up) is different: a < b < c would usually look like if a < b < c: and be read "If a is less than b is less than c...".

Spitemaster
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  • Shouldn't it be "a is less than b is less than c"? – Mitch Feb 13 '20 at 18:25
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    @Mitch No. That's not the common usage. – Spitemaster Feb 13 '20 at 18:26
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    Leaving out the 'is' is much less common, and unnatural sounding. – Mitch Feb 13 '20 at 18:30
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    @Mitch Not in higher math, which I was speaking to. In my experience, at least. – Spitemaster Feb 13 '20 at 18:39
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    My experience is that both phraseologies appear with reasonable frequency in higher math, with the is included in all other cases, including basic- and intermediate- level math. – Ryan Jensen Feb 13 '20 at 21:00
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    When you write on the blackboard teaching class, you say just like this. – Matsmath Feb 13 '20 at 21:11
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    @Spitemaster not in any math, higher or lower, I've seen. Of course it all depends on context...under what circumstances do you see 'a<b<c'? I can only think of the 'If a<b<c, then...' context. – Mitch Feb 13 '20 at 21:13
  • @Mitch For example, one could write b<a: *result 1*, a<b<c: *result 2*, c<b: *result 3*. – Spitemaster Feb 13 '20 at 21:20
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    -1 Disagree in strong terms. The suggested translation isn't a grammatically correct English statement; and most critically, it loses sight of the fact that relations are the verbs of the algebraic language, and really must include the to-be "is" for clarity and correctness. I consider this to be so important there's a first-day handout in the discrete mathematics class I teach on exactly this issue. – Daniel R. Collins Feb 14 '20 at 01:50
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There is many different way to say that. But, I think this is clearest way : "A is less than B and B is less than C"

Ali
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5

I would just take it at face-value and read it left-to-right: "A is less than B is less than C."

dev_willis
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2

I would say b is between a and c non-inclusive or, in the middle of a sentence a less than b less than c. (Yes, in a mathematical context I would not use the is.)

Mike Graham
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    A matter of opinion, Mike. For me "<" means "is less than" in most cases, it makes for better English when you are reading the expression. The only cases where I wouldn't use "is less than" are when there is an element of assignment; for example "Let a be < b" or, preferably "Set a < b" which would be pronounced "Let a be less than b" and "Set a to be less than b". If I can't read algebra as normal English I can't follow it when it gets complicated. – BoldBen Feb 13 '20 at 06:42
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    Why not “strictly,” instead of “non-inclusive” – cole Feb 13 '20 at 19:45
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    This does not imply that c is greater than a, and thus does not capture the spirit of the original equation. It only says that b is between two other values, but we don't know which of those other two are larger. – Brian R Feb 13 '20 at 19:54
  • @cole You certainly could -- I'm simply conveying what I, someone who has done a decent bit of math, would personally say. With strictly, I would have to say it twice, which ends up burying the lead. – Mike Graham Feb 14 '20 at 00:17
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Like all writing, I think this depends on who you expect to be the reader. If it were a purely math audience, you would not bother to use words.

1) a < b < c

would be enough.

If it a non-math audience, and the context was already established that a was less than c, then "b is between a and c but doesn't equal either" would be pretty clear.

If the audience was more visual than verbal, you could draw a picture of a, b, and c appropriate for the problem space.

Everything about words depends the writer anticipating the interpretation by the reader, without becoming too words or pedantic.

Simple sentences to convey simple ideas. Sentence fragments, even.

cmm
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  • While a good answer for a written style, this is bad for a verbal or spoken style, such as a teacher talking to a class – Chronocidal Feb 17 '20 at 12:54
  • @chronocidal, your comment is spot on. Checking back to the question, the OP asked about "is there a better way to say". I took that to be a an informal word for "write", but he wrote "say". – cmm Feb 17 '20 at 13:12
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In fact, a more mathematically correct way to say that would be this:

a is strictly less than b and b is strictly less than c

The < symbol doesn't denote just inequity, but strict inequality.

Michael Rybkin
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    Inequity is strict unless specified otherwise. This answer makes a statement which is already, of necessity, wordy, even more complex for no convincing reason. – Ryan Jensen Feb 13 '20 at 21:02
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    @RyanJensen I know that this is an old comment but: this is not necessarily true. A lot of analysts will use "less than" to mean the weak inequality. Indeed, Barry Simon makes this explicit in his five volume treatise on mathematical analysis. In most contexts, this distinction doesn't make much difference, but if the distinction does matter, then the answer here is spot on. –  Nov 18 '21 at 12:13
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I would say "b is strictly between a and c", or if it's very clear from context or the distinction with a ≤ b ≤ c doesn't make a difference (e.g. in case of a continuous probability) just "b is between a and c". I would consider the latter way of saying it much more conventional however.

Tomlish
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    However "b is (strictly) between a and c" doesn't tell you anything about the relationship between a and c (i.e. a < c). – KillingTime Feb 13 '20 at 13:05
  • I'd say it's quite customary that a is the smaller and c the larger value when saying "between a and c" – Tomlish Feb 13 '20 at 13:13
  • Also in non-mathematical speech, e.g. "between 10 and 12 grams of flour" or "I'll be there between 8:30 and 9" – Tomlish Feb 13 '20 at 13:16
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    If the equation was a>b>c, how many people would say that "b is between a and c" isn't true? – KillingTime Feb 13 '20 at 14:52
  • I can't imagine a context where OP's statement isn't intended to be strictly true mathematically. This suggestion says something that's not equivalent. – jimm101 Feb 13 '20 at 16:26
  • Do I really deserve a downvote? Like in my example, it occasionally doesn't matter whether the boundaries are included or not. In that case it's not strange to be imprecise about them in speech either. Regarding a>b>c, it is pragmatically often a bit odd to write this equation. A reason for this is that we commonly order from small to large, both in everyday speech and in mathematics itself, e.g. in the standard notation for intervals. – Tomlish Feb 13 '20 at 17:27
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Taking your second quote, which lists both a and c with respect to b, you can rearrange it slightly:

a is less than b, and c is greater

Chronocidal
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  • so is c greater than a or b? – Kevin Feb 13 '20 at 21:31
  • @Kevin Yes. However, due to the way that the English language functions, the implicit object of the second part of the compound sentence is the same object as the first part of the compound sentence: in this case, b (with both a and c being the subjects). We do not need to repeat "than b" - that would be redundant, and thus bad grammar – Chronocidal Feb 14 '20 at 00:40