Number notation relates to forms of notation where numbers are used to designate individual notes. It has the advantage of using symbols that are already familiar and is useful when dealing with instruments where numbers are written on the notes, such as some tuned percussion instruments. It can also help you to understand the degrees of the scale and the relationships between them, and to this end is sometimes used for vocal warm-up exercises. However, it has limitations - you need additional symbols to indicate note lengths, metre, etc.; it doesn't give a visual indication of pitch, although can be adapted to do so; and it works best when applied to scales of no more than 8 notes.
A simple number notation would notate the melody "Merrily we roll along" like this:
3212333 222 355 3212333 22321
Additional symbols could be used to clarify the rhythm:
Forms of number notation are quite widely used, for instance in East Asia, and some examples are given below. In addition, there is the use of numbers in figured bass notation of the baroque period, and some European keyboard tablatures; while Roman numerals are often used for writing down chords.
These examples are from a book published in Hong Kong. They add some additional information and symbols from western notation - time signature, bar lines, dots and connecting bars to indicate quavers and semiquavers.
Other Hong Kong publications use numbers alongside western stave notation:
Number notation is also used to notate western popular songs, in which case they might act as a supplement to the player's aural knowledge of the tune.