Don't undestand this particular use of ない form


This is the example sentence from one of my textbooks.

時間があるから、昼ごはんを食べて行かない？

Textbooks translation - "We have some time so how about we grab a bite to eat
and then go?"

I don't understand why the ない form here doesn't translate to a present
negative like 'grab a bite to eat and don't go" hence my confusion over its
meaning. Either the translation is somehow off (unlikely), this sentence is
simply incorrect (unlikely) or theres some rule I'm forgetting or don't know
about (very much likely). Please help.




The textbook translation is correct, but kind of roundabout. It's more like,
"We have some time, so **_why don't we_** grab a bite to eat and then go?" The
"ikanai" would be pronounced with a rising intonation that affords a
suggestion: `[行かない？]{LLLH}`

It does make sense if you do some "logical shifts" of the **not** part of the
meaning and some semantic substitution.

> ...昼ごはんを食べて行かない  
>  → Will we eat and then **not** go?  
>  → Will we **not** eat and then go?  
>  → Will **not** we eat and then go? (ungrammatical and/or strange in
> English, but added for visual reference)  
>  → Wo ** _n't_** we eat and then go?  
>  → "Wo ** _n't_** we eat and then go?" + rising-intonation = "Why do **
> _n't_** we eat and then go?"




The textbook translation is correct, but kind of roundabout. It's more like,
"We have some time, so **_why don't we_** grab a bite to eat and then go?" The
"ikanai" would be pronounced with a rising intonation that affords a
suggestion: `[行かない？]{LLLH}`

It does make sense if you do some "logical shifts" of the **not** part of the
meaning and some semantic substitution.

> ...昼ごはんを食べて行かない  
>  → Will we eat and then **not** go?  
>  → Will we **not** eat and then go?  
>  → Will **not** we eat and then go? (ungrammatical and/or strange in
> English, but added for visual reference)  
>  → Wo ** _n't_** we eat and then go?  
>  → "Wo ** _n't_** we eat and then go?" + rising-intonation = "Why do **
> _n't_** we eat and then go?"

