Take for instance the number 64. How is it that it can represent a line with a length of sixty-four units, a square with one side the length of eight units and a cube with one side the length of four units?
I get that there are three different units of measurement involved; say, cm, cm2 and cm3. However, it doesn't seem to fully address my question.
I'm currently reading Hung-Hsi Wu's Pre-Algebra to hone in my definitions before moving on to calculus and it struck me as odd that the author explained addition, subtraction and division of two fractions strictly on a one-dimensional number line but when it came to the multiplication of two fractions, he resorted to a two-dimensional plane with each side represented by one of the fractions, which got me to the question above (the two are not really related, but I want you to know the mental framework I'm currently in).
The author defines a fraction as a point on a number line which is the definition of a number, too, and I am unable to grasp how something that represents a point on a number line can also be utilized to represent an area or a volume (I can understand a fraction as representing the distance between two points on a one-dimensional line as the point on which the bigger one of the two numbers would land had the two numbers been pulled back such that the smaller number was placed on the origin.)
