Most Helpful Girl
The real question is why do you have her number and what are the intentions? And the answer is NO. Speak when you desire to speak. If you have something to say, say it. If your concerned that you will lose this person, then either let them know and ask how they would like to be communicated, or stop getting strangers numbers. It depends on what type of person you are and who you're talking to. You two needs to set the rules you value in this as well as boundaries.
Most Helpful Guy
Depends on the girl and the strenght of your emotional connection to her.
My ex would almost always text me first when were dating. But at the start I was initiating 7/10 times. As she got to know me she wanted to text me first as she wanted to speak to me more oftem or just wanted my attention.
If you are constantly initiating and recieving careless responses (or none at all) it's time to stop texting "X" girl.
If you're initiating and getting solid dates from "X" girl, then yes. Continue this behaviour with "X". This method is having success with her.
It's all about evaluating your progression with a girl over a certain period of time (usually 14-28 days) and then making decisions depending on the outcome of your evaluation.
If there is NO progression being made. Cut all contact. No point in wasting YOUR valuable time on someone who doesn't want to include you in THEIR life.
I hope this was in some way helpful.
Best of luck for the New Year!