Fractions rational and decimal, and wiggle

Consider the number two and three-quarters.  We can write it in mixed rational form, `2\frac{3}{4},`  or in pure rational form, `\frac{11}{4},` or in decimal form, `2.75`.  Each of these forms can be modelled in wiggle notation :

`\qquad\qquad\qquad 2\frac{3}{4}  =  2;_{4}3,`

`\qquad\qquad\qquad \frac{11}{4}  =  0;_{4}11,`

`\qquad\qquad\qquad 2.75  =  2;_{10}7_{10}5`.

Rational forms ("fractions") and decimal forms ("decimals") are two alternative ways of representing the numbers that live between the whole numbers.

Fractions and decimals are alternative notations for rational numbers.  Although they are used to represent the same things, the notations themselves are usually taught as unrelated.  Wiggle notation serves as a kind of bridge.  As we shall eventually see, wiggle glues a lot of things together.