Why an .numbers file won't open on iPhone
When someone saves a spreadsheet in Apple Numbers, it gets an .numbers extension - Apple's proprietary format, the iWork cousin of Microsoft Excel. It sounds odd, but an iPhone without the free Pages/Numbers/Keynote apps installed will refuse the file, and people often don't want to install three Apple apps just to read one attachment.
Nothing is actually broken. The author saved it on a Mac, iPad or iPhone where it opens perfectly, and forgot that on iPhone the file is a locked box. People often assume it's corrupted or that the sender made a mistake, when really iPhone simply doesn't speak Apple's format.
What an .numbers file actually is
Here's the trick that makes it easy: an .numbers file is really a ZIP archive in disguise. Inside, Apple Numbers stores its document data plus a ready-to-view QuickLook PDF preview, so Finder and Mail can show a snapshot without launching the full app. That embedded PDF is the key to viewing the spreadsheet anywhere.
You could prove it by renaming the file from MyFile.numbers to MyFile.zip and digging out the preview PDF by hand - but on iPhone that's fiddly and easy to get wrong. The tool on this page does it for you automatically: it opens the ZIP in your browser, finds the PDF, and shows it, with no renaming and no guesswork.
The fastest way: open it right here on iPhone
The tool at the top of this page is the quickest route for a iPhone user. Drop the .numbers file in and it is unzipped locally by JavaScript, the embedded PDF is extracted, and the spreadsheet appears on screen. Nothing to install, no account, and crucially nothing is uploaded - the file is read entirely inside your browser, so even confidential spreadsheets stay on your iPhone or iPad.
From there you can download a PDF copy. That single step solves the problem permanently: a PDF opens in every app, so you never have to wrestle with the .numbers extension again. It also works where you're not allowed to install software, which is a common reason iPhone users get stuck.
Other ways to open .numbers on iPhone (and their downsides)
iCloud.com is a free option if you have an Apple ID: upload the file to iCloud Drive, open it in Apple Numbers for iCloud, and read or export it. The downsides are creating or signing into an Apple account and uploading your private spreadsheet to Apple's servers, which is slower and less private than a local tool on iPhone.
Online converters like Zamzar or CloudConvert can turn .numbers into PDF or Excel format, but again you upload your file to a third-party cloud, often hit a free-conversion limit, and may wait in a queue. For a one-off iPhone user, opening it right here is simply the path of least resistance.