I can't remember when and where I had this discussion, but I remember being corrected when I was speaking by a stranger saying that it is never correct to say give me half of this; instead, the grammatically correct phrase would be give me one half of this. I've never been a pro at where numbers fit in with the English language, so maybe someone here could shed some light on this.
It is perfectly acceptable to say "give me half of that". In English, "half" in understood on its own to mean "one of two equal parts of something".
To put it another way:
- It would make no sense to say "give me no halves of that". You would just say "give me none of that".
- It would make no sense to say "give me two halves of that". You would just say "give me all of that".
Saying "give me one half of that" is redundant. It's equivalent to saying "give me one of one of those two equal parts of that."
Comment posted as answer - as requested
In idiomatic usage, you would seldom say 'one half of this'. You might say 'give me one half-pound pack of sugar' but the hyphen shows that it is a different construct. You might say 'give me one third of that' (as opposed to 'two thirds of that'), but with halves, the alternatives are none and all. However, even with thirds, it would be more usual to say 'a third' than 'one third'. So, whoever 'corrected' you was actually misleading you.
I had this situation recently. It was so embarrassing when the saleslady told her co-workers about what I said. I asked her to give me one half kilo (1/2 kilo) of prawns and 3 pieces of fish. She ended up giving me one and one half kilo of prawns. When she asked for the payments and I asked the amount, i freaked out. I was not expecting that she actually gave me 1 1/2 kilo of prawns. I told her I asked for one half not one and one half kilo. She got angry and her co-workers started to laugh. They were all telling me I was wrong. They said it's half not one half. In my understanding, "one half" and "half" are both correct it depends on how you use it with other supporting words.
You don't need the 'one' in expressions like 'give me half a cookie'. Where you do need the 'one' is in when units of measurement are involved, like "please give me one half pound of sugar". You can use 'a' instead of 'one', but leaving out any determiner is wrong.
Give me one half pound of sugar.
Give me a half pound of sugar.
Give me half a pound of sugar.
*Give me half pound of sugar.
*Give me one half a pound of sugar.
?Give me a half a pound of sugar.
(You do hear the last quite a bit in the U.S., although not in England; this came up in another question here.)