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The phrase "a basic mistake in" is correct and usable in written English.
It can be used when identifying a fundamental error within a specific context or subject matter.
Example: "There was a basic mistake in the calculations that led to the incorrect results."
Alternatives: "a fundamental error in" or "a simple error in".
Exact(4)
But Frieden acknowledges a basic mistake in his communication efforts.
If you took that line seriously, John, you are making a basic mistake in comprehension.
"If you exhaust all your synonyms, you are making a basic mistake in style," says John Orford.
This might have protected what appears to be a basic mistake in how Samsung integrated SwiftKey on their devices.
Similar(56)
That was a basic mistake back in 1972.
In the end it was a sad, basic mistake in the field which proved the turning point.
Independent experts said they were surprised that Boeing's inspectors had not caught such a basic mistake when the plane was built in the mid-1990s.
He describes how a basic mistake about the federal debt ended up in the state-of-the-union address without being checked because it buttressed the case for tax cuts.
Indeed, in his later writings he seemed on the point of saying that a basic mistake might have been made somewhere along the line.
"They've been letting taxes determine their strategy, and that's a basic mistake.
"We made a lot of basic mistakes in Palma - racing mistakes - and the reason we're racing now is to iron these out, get savvier, get sharper," said Percy recently.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com