When I ask a women for her number (but you should do this with everybody) I will always hand them over my cell phone. This is great in many ways, allow me to explain. For One, you minimize the chance for rejection because it's more of a demand rather than a question. They can say no if they want but at least you don't give them the option of saying no by asking a yes or no question. Two, If you forgot the girls name she will go ahead and type in your phone for you. You have no idea how many times I forget a persons name and I'm saved because they type it in, phew! Three! You look trustworthy because your handing them something of value and you are trusting them with it. So it's an easy way to build some sense of trust. Usually when then have my phone I will turn away and not keep an eye on them, while still maintaining a conversation if any. Four, it makes things much less awkward, you don't have make them tell you their phone number and have repeat each number as they go along, that's just awkward in my opinion. And Five, if you have a flashy phone then extra points are won!

Edit: A couple sedditors have given a couple other good reasons why this is good. Six, If you have an interesting background such as a band, a landscape, a sports team etc. it could potentially lead to a conversation if she happens to have the same interests, you never know. Seven, and this one is crucial, you always text immediately right in front of her and say "I'll text you something so you can have my number" this way you can verify the validity of the phone number and at the same time send a short witty response with your name. Always try and relate that witty text to something related to what you spoke about.

Also, someone said "well what if some other girl texts me as she's putting the number on my phone?' Nothing wrong with that! In fact it's a good thing because she will see that other women are interested in talking to you and hence give you some sort of social acceptance. Women want to know that you are accepted by other women. It gives them a sense of security and they feel more comfortable knowing that other women feel comfortable around you. One girl on the thread said it would "instill some healthy competition".

It's just one of those things that makes a difference. Of course, this is only my opinion on this you are welcome to disagree and share your method as well.