"He arrived at approximately 5pm."

"At" means the exact time. "Approximately" means near that time. Therefore, isn't "at approximately" grammatically incorrect? Isn't it a contradiction? But how else can one re-write the sentence? Leaving out "at" just doesn't sound correct.

Thanks for your time.
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The phrase "at approximately" sounds OK in your sentence, though I see what you're driving at.

Here is a possible paraphrase:

He arrived around 5 p.m.

However, don't be surprised if you hear this:

He arrived at around 5 p.m.

This is more casual English. Some may even say it is wrong, so I would avoid it in more formal writing.

Aha. Thanks for clearing that up for me. Further, I once read that "at roughly" is acceptable, whereas "at approximately" is wrong. I can't remember the source of this information but it was from a grammarian. I don't see why they both shouldn't be wrong.
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"At roughly" is acceptable informally, whereas "at approximately" is acceptable formally.
Mister Micawber: How can "at approximately" be accepted formally when the words contradict themselves? I'm confused. Please respond.
At approximately / at about / at roughly

These are synonymous and there is no contradiction in any of them that I can see. The speaker specifies a range of time to meet at.
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Mister Micawber: At -- Preposition expressing location or arrival in a particular place or position
Approximately -- adverb: close to; around; roughly or in the region of

How can the words be synonymous with the definitions I copied from the web's dictionary?

Thank you.
You are just repeating yourself. I do not believe the definition of 'at' indicates a point place or position.
Anonymous"At" means the exact time.
No it doesn't. Where did you get that? "at" is a preposition used to connect a verb with a time or place. Neither the time nor the place have to be exact in order to sanction the use of "at".

arrive ___ time: He arrived _at__ [ 3pm / roughly 4pm / approximately 6am / exactly 8am / a time between 3pm and 4pm].
arrive ___ place: He arrived __at_ [the station / the outskirts of the city / a place somewhere between the bank and City Hall].

There is no more contradiction in 'at approximately 3pm' than there is redundancy in 'at exactly 3pm'.

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