Write in your book!
Unless your Shakespeare play book is a very expensive, specially bound work of art that cost hundreds or thousands of dollars, or maybe you read from a 1623 First Folio, or—I can't think of any other excuse—you need to write in that thing.
As it is, the book you are reading is probably inexpensive and there are probably bazillions of the exact same book in the world. It's an Arden, you say? Or your Bevington Collected Works? Or even a Riverside with its cheap pages and awful typography? Write in it. By writing in your Shakespeare book, you change a bundle of wood pulp and printing ink and bookbinder's glue into a treasure.
Imagine a 1902 copy of Shakespeare's plays with immaculate pages and probably a loose binding, one of the thousands of editions available at the time. Now imagine it with your grandmother's notes in it, what she thought of Desdemona, how annoyed she was with Falstaff, her tear stains in Titus. Which book is more precious?
But don't take it from me. Read the essays on the left by Steve Leveen and Mortimer Adler, who are much more important than me.