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could someone tell me how to read this mathematical expression aloud?

exp⁡(√(cos⁡x )+1-x^4)

I didn't take math in school so I would have no clue... I just need to be able to read it aloud.

Thank you very much!

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  • I just need to read it aloud, like a math teacher would mumble as they write this down on a board. It is for a specific reason - and no I'm not working with a blind mathematician (they wouldn't need anyone's help like this). I can read this aloud in Japanese with my limited math knowledge but not in English as I only took math in Japanese.
    – user18488
    Sep 17 at 17:52
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    As a maths teacher, I wouldn't read it as writing it. Because I know that is confusing for students. I'd write it and say "this expression".
    – James K
    Sep 17 at 17:56
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    Well, I can read this aloud in Japanese which is very simple - イーのエックス乗 括弧 ルート 括弧 コサインエックス 括弧閉じる 足す 一 引く エックスの四乗 括弧閉じる. If a math teacher read this aloud in a Japanese high school math class, students would have no trouble dictating it and writing this down on their note pads. You should be able to do that in English?
    – user18488
    Sep 17 at 18:04
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    Also, I may be wrong but this seems to be "exp(x) √(cos⁡x).... and not exp(√(cos⁡x )...)
    – James K
    Sep 17 at 19:28
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    @user18488 A maths teacher, if they try to say it at all, will probably just recite the symbols they're writing without too much concern for whether what they're saying would be meaningful without the accompanying visual representation. Like one might say "x to the power y plus 1", but that doesn't tell you whether it's x^(y+1) or x^y+1. Even for something so simple, it's already quite clunky to clarify which one you mean.
    – NotThatGuy
    Sep 19 at 15:34

10 Answers 10

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There is no single correct answer.

For example, I'd read "exp" as "e-to-the power". You might use "square root" or just "root" (since square roots are the most common)

Moreover, if you are reading aloud, you are probably doing something wrong. Spoken maths is normally ambiguous, and the whole point of maths is to be unambiguous. There is a reason for writing things down.

However if you are writing this and reading it as you write it you could say:

e to the {pause} root-of-koz x {short pause} plus one {short pause} minus x to the power four

And I know that most students would find my speaking rather annoying (as it interferes with their note taking) This why you shouldn't speak maths out loud.

On the other hand, if you are talking over the phone, you can be more or less completely unambiguous by spelling out the brackets

"e to the power of ... open bracket ... the square root of ... open bracket ... the cosine of x ... close bracket... plus one ... minus x to the power four ... close bracket.

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  • I've edited based on your comments
    – James K
    Sep 17 at 18:01
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    Ah! I see now I understand what you mean by "unambiguous" - I think this is what I was looking for. Thank you very much!
    – user18488
    Sep 17 at 18:09
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    Nice detail the pauses, long and short, to convey the hierarchy of groupings in the formula.
    – Pablo H
    Sep 19 at 13:36
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    "Spoken maths is normally ambiguous" - not only that, but it's also really hard to follow for anything beyond the simplest of formulas.
    – NotThatGuy
    Sep 19 at 15:27
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    I don’t know if this is a US/UK thing, but when I earned my math degree in the US, I never heard “power four”, just “to the fourth”. As in, “minus x to the fourth”. Sep 19 at 20:15
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Unfortunately, IMHO, apart from James K's telephonic version, the answers given so far are wrong or, to be generous, ambiguous. The problem lies in exactly what is contained in the square root. Is it just cos(x) or is it all of cos(x)+1-x^4 ? The OP's written version where cos(x) is inside parenthesis sqrt(cos(x)) is clear, it is only the cos(x), the rest +1-x^4 is not inside the root. However, if you don't make that distinction when reading it, transcription errors could occur.
Thus the correct version, without repeating it, is James K's telephonic one.

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Context: I am a professional mathematician and speak British English.

I would say something like

exp of, all in brackets, root cos ex plus one minus ex to the four

Here "cos" has a hard c, short o and voiced s (like the second syllable of "because"). I am surprised that some other answers say "cosine" in full, as I only ever hear this in the specific phrase "the cosine rule". I wouldn't bother specifying that "root" means "square root", or that it only applies to the immediately following term, both of which I would expect to be understood - if I meant something different I would specify, e.g. "cube root open bracket cos x plus one close bracket".

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  • I am a professional mathematician and speak British English and I hear both cos and cosine about equally often, also when not followed by the word rule. I know our kind does not get out much, but, really?
    – Deipatrous
    Sep 19 at 9:29
  • I would have considerable difficulty working out from that where the brackets are meant to go. Is it root(cos(x))+1, or root(cos(x+1)) or root(cos(x)+1)? Sep 19 at 16:27
  • @MichaelKay The first one, since either of the others would need brackets, and I didn't mention any. I would never write cos(x)+1 with brackets, so why would I say them? Sep 19 at 18:41
  • This matches my US English math experience except I’ve heard the full “square root” often and we say “to the fourth” instead of “to the four”. Oh wait, also we would say “e to the” instead of “exp” and obviously “parenthesis” instead of “bracket” Sep 19 at 20:17
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    @ToddWilcox In general I say "e to the" more often than "exp of", but for me they indicate different ways of writing the same function. Here it was written in exp(...) form so that it is how I would say it. 2 days ago
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You can read it as:—

Exponential of the square root of (cosine of x plus 1 minus x to the power of 4).

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    How do you say "("
    – James K
    Sep 17 at 16:49
  • @JamesK You don't pronounce the parenthesis. In mathematics, parenthesis denote modifications to the normal order of operations. You would not adjust the order of speaking it.
    – Astralbee
    Sep 17 at 17:19
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    @Astralbee you can pronounce it as "open bracket" as in a few of other answers. The point of the question is how to pronounce things.
    – justhalf
    Sep 19 at 9:26
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    @Astralbee The question is about how to say it. The parens don't help the OP know how to say that part of the expression, so an answer should either use words for the parens, or leave them out of the answer if they're unpronounced.
    – gotube
    2 days ago
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    I agree with you, no doubt about that. I'm just weirded out that people seem to oppose the question on how to pronounce certain symbols. This is not Math Educator stack where presumably it's useful to help teach the OP that this is not the way it should be done. It's just asking how to pronounce things. I didn't find this question weird at all.
    – justhalf
    2 days ago
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American who got my degree in the U.S. If I were being more formal, I would say, typically after writing it on the board,

e to the quantity: the square root of cosine x, plus one, minus x to the fourth.

The colon would be a long pause and the commas I wrote between the terms would be short or no pauses, so as not to confuse the listener into thinking that I’m ending the quantity in parentheses. Variations I might expect to hear, or even use:

  • “exp of,” “exponential of” or “e to the power of” instead of “e to”
  • “square root of cosine x” or “root cosine x”
  • “x four,” “x to the power of four,” “x to the fourth power.”

I’d just about always say “the quantity” when grouping terms in parentheses or brackets, unless there’s no chance of being misunderstood. This isn’t a great way to unambiguously identify which terms are being grouped, and I’d be prepared to write it out. But people would normally infer that I didn’t mean cos (x+1), or I would have said, “cosine of the quantity x plus one,” and I didn’t mean exp(cos(x)) + 1, or I’d have said, “e to the cosine of x, plus one.”

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Language is always about agreement between two or more people to use a certain grammar and vocabulary to convey some meaning. There is absolutely no issue with agreeing to a certain kind of grammar and vocabulary such that the mathematical expression you desire to convey can be read out very quickly and unambiguously interpreted as follows:

exponential open square-root cos x plus 1 minus x to the power of 4 close

This is far more succinct than James K's version, and is actually used in conversation between some mathematicians. Let me emphasize that you of course need to agree on the language for it to work.

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    Contrary to another answerer's claim, this is not ambiguous because one has to first agree on precedence rules in order to make reading mathematical expressions out sufficiently usable in the first place. One cannot claim that there is ambiguity in what the square-root is applied to, otherwise one should also claim that there is ambiguity in what the "to the power of 4" applies to (which the other answerer didn't). Why not (1−x)^4? Because of precedence rules.
    – user21820
    Sep 18 at 7:47
  • I'm not aware of any standard rule determining whether the phrase "the square root of" (or its abbreviated form, "square root") should include or exclude any following added or subtracted terms. I think I would say that this spoken phrase is unambiguous, but only if the speaker and the listener have agreed that the square root excludes following terms. Sep 18 at 14:32
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    @user21820 I agree and disagree. I conceede I overlooked the additional ambiguity concerning 1-x^4 versus (1-x)^4 so this is still ambiguous. I disagree that there is any universaly agreed precedence rule in spoken mathematics. In the best part of 70 years of maths, electrical engineering and software engineering I have never come accross one, which is why we always write it down. To make your version as unambiguous as possible you could say e to the power open square-root open cos x close open plus one close open minus x to the power four close close Sep 18 at 23:34
  • @PeterJennings: I didn't claim there are universally agreed precedence rules! I stated very explicitly "no issue with agreeing ... such that ..." and that this particular grammar involved in my example is used among some mathematicians. But you must agree on precedence rules otherwise you are actually not doing mathematics like almost all mathematicians do. For example, all mathematicians write "1+4x+3x^2" and never "(1+(4x))+(3(x^2))" (it's just stupid). Precedence rules are meant to facilitate efficient communication.
    – user21820
    Sep 19 at 5:58
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    @user21820 Again, I agree that you would never write it "(1+(4x))+(3(x^2))" because the simpler written forn is adequate and unambiguous. But when spoken you cannot hear the brackets if they are omitted. Sometimes this doesn't matter as "one plus 4x plus 3 x squared" is almost unambiguous (even this could be mistranscribed (1+ 3x+4x)^2, but is highly unlikely in, say, the context of quadratic equatins). As we are all apparently agreed that there is no universal spoken method, I maintain that, unless you are exceptionally pedantic, the possibility of mistranscription remains. Sep 19 at 16:41
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ee raised to a sum containing three terms, these being one, the square root of the cosine of eggs, and the product of minus one and eggs to the fourth

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e to the power of cosine x plus one minus x to the fourth.

Say it a bit fast, no pauses. Like this: e to the power of the square root of cosine x-plus one-minus-x-to-the-fourth.

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  • 1
    Why did you post two answers here? Also this one doesn't have a square root in the first sentence.
    – Laurel
    Sep 19 at 17:05
  • @Laurel not sure which one is better.
    – BigMistake
    Sep 20 at 0:06
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    While a case could be made for posting them separately, it might be better in this case to put them into a single answer, explaining that you might use either and covering the strengths and weaknesses of each.
    – Laurel
    2 days ago
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If you weren't writing it and only speaking verbally, e to the power of the square root of cosine x, 1, and negative x to the fourth.

Saying and implies they are all in one group together under the square root.

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If I knew I would have to read this out loud then I would rearrange it first to reduce ambiguity.

e to all of one minus the quantity x to the fourth end quantity plus the square root of cosine x.

All of is useful for situations where the parentheses should encompass the largest possible section, like here.

The quantity is useful for denoting parentheses, since we still have some ambiguity even after rearranging.

Another nice option is if parts of this have meaning you can specify it in parts...

e to the foobar where foobar is foo plus one minus baz, with foo being the square root of cosine x and baz being x to the fourth.

I would only do that if you had sensible names for these things, though, like if the equation represents something in physics or economics, etc.

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