How are symbols evaluated

Suppose we start with an undefined symbol

We can assign a value to this symbol using = (Set)

How does Mathematica know to replace every instance of x with its value?

It uses a global look-up table. We can find the rules for evaluating symbols by calling OwnValues

The symbol :→ (typed :>) means replace the lhs with the rhs, and only then evaluate the rhs

HoldPattern is a way of expressing a rule for a symbol without having the symbol transformed by other existing rules

Since we used SetDelayed, we see that in this case y is replaced by a new random number each time

How functions are evaluated

Recall our Fibonacci sequence

fib has no OwnValues. That’s because if we evaluate

no rule is applied. Rules are only applied if we give an argument to fib

These kinds of rules (that require arguments) are known as DownValues

The order of these rules shows the order in which Mathematica will attempt to apply them

There are also UpValues and SubValues, but they are a bit obscure...

Understanding function evaluation

Trace helps us understand the order in which Mathematica evaluates expressions

Trace will not show the inner workings of all functions

However, FindRoot supports EvaluationMonitor