I suffer from constant Night Terrors and I'll tell you what happens when you don't sleep.
Day 1: Groggyness, unfocused
Day 2: Groggyness, more unfocused, constant hunger, cold and heat flashes.
Day 3: Extreme fatigue, completely unfocused, possible fever, cold and heat flashes, no hunger, weakness.
Day 4: This marks the point of Micro Sleeping (partial sleep that lasts less than a minute, confussing dreams with reality.) Anxiety, fever, extreme weakness. By now your just laying there.
Day 5: Welcome to Hallucination Centural! On top of feeling every thing from day four, your going to start hearing things. Scratching on the wall, knocking, screams, I even heard a lion roar once!
Day 5 1/2: 6 Hours later the vision part of the Hallucinations will kick in. Shadows are going to dart around, you'll see people, animals, buggs, it's not limited.
Day 5 3/4: 3 or 4 hours later your going to start feeling the hallucinations.
Day 6: Hallucinations contenue and your gonna feel worse.
Day 7: You need a doctor if you don't get one then your probally not going to last much longer. They'll put you to sleep and your going to have Lucid Dreams were you wake up in your dream.
That's as far as I'll go because I don't remember much from when I went 11 day without sleep.
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According to several world records, you would die after about 11 days awake, but it is extremely unlikely that you would be able to stay awake that long, since your brain commands you to sleep.
But, to my knowledge, what happens when you SLEEPis your brain cells are fixing themselves. Also trying to recap all you learned that day and put it into a "storage" for later use. When you DON'T SLEEP you pretty much do the opposite. Your brain is commanding you to sleep, so you can repair your brain cells. That is why it is not okay to study all night!
You would die. Sleep is not an optional function. People need REM sleep to survive. If you don't sleep, the neurons in your brain can't recharge. After a few days without sleep, you'd start experiencing some pretty severe side effects like paranoia, fatigue, and hallucinations. The longest anyone's ever lived without sleep is, I think 11 days. Sleep Deprivation is really not fun. It's even been used as a torture.
Basically, if you never slept, you wouldn't live very long.
You would start vomiting blood and your head would hurt and you would be very weak and then you would die.