# Using Let to Create Local Variables

Local variables are variables that only exist within a local environment. Here's an example:

(define (foo x)
(define a 5)
(+ x a))


The local environment is the environment created by the function foo, and the local variable is a. Note that x is not a local variable, even though it also cannot be accessed outside of foo—it is formally called the parameter.

## Introduction to let

The special form let is essentially a call to a lambda function, arranged differently. For example, take the following lambda function call:

-> ((lambda (x y z) (+ x y x z)) 1 2 3)
7


This is equivalent to the following let statement:

-> (let ((x 1) (y 2) (z 3)) (+ x y x z))
7


When will this ever be useful? Two words: local variables. Rarely will we use a let statement to simply call a lambda function. Instead, we use it create local variables inside of a function.

## An Example: Polynomials

Let's say we want to use Racket to compute the following polynomial with any given x and y:

$$f(x,y) = x(1+xy)^2 + y (1-y) + (1+xy)(1-y)$$

Rewriting this ugly polynomial as an ugly procedure:

(define (f x y)
(+ (* x (+ 1 (square (* x y)))) (* y (- 1 y)) (* (+ 1 (* x y)) (- 1 y))))


Yuck. Instead, we could use some substitution:

$$\displaystyle a = 1 + xy$$

$$\displaystyle b = 1 -y$$

So that we get:

$$\displaystyle f(x,y) = xa^2 + yb + ab$$

Okay, I guess that's better. Writing this in Racket, we will define a helper function called f-helper so that we can use substitution:

(define (f x y)
(define (f-helper a b)
(+ (* x (square a))
(* y b)
(* a b)))
(f-helper (+ 1 (* x y))
(- 1 y)))


Take a minute to confirm that this does the same thing as the earlier definition of f. As we learned in the previous section, we don't really need an extra function definition inside f. Instead, we can use a lambda:

(define (f x y)
((lambda (a b)
(+ (* x (square a))
(* y b)
(* a b)))
(+ 1 (* x y))
(- 1 y)))


Sadly, even after all this substitution and reorganizing, it's still a bit messy. This is where let comes in!

(define (f x y)
(let ((a (+ 1 (* x y)))
(b (- 1 y)))
(+ (* x (square a)) (* y b) (* a b))))


Finally, we get a more readable version of our initial polynomial function f. We can clearly see that we're assigning a value to a and b, then plugging it into the body of the let statement.

## let: General Form

The general structure of a let statement is

(let ((<var1> <exp1>)
(<var2> <exp2>)
...
(<varn> <expn>))
<body>)


Remember, underneath, this is nothing more than a lambda call. The above structure is equivalent to

((lambda (<var1> <var2> ... <varn>) <body>)
<exp1> <exp2> ... <expn> )


Try out these expressions (and more!) in the Racket interpreter.

(Note: A semicolon denotes a comment. Racket will ignore the rest of the line after a semicolon.)

(define y 10)

(let ((y 0)) y) ;; notice that let overrides global vars

(let ((x 10)
(z x))
z) ;; this will break, translate to lambda to see why

(let ((a 1))
(let ((a 2))
(let ((a 3))
a))) ;; nested lets are valid.

(let ((test 'wait-what?))
5)
test  ;; let only binds variables inside its body

(let ((a 1))
(+ a (let ((a 2))
(+ a (let ((a 3))
a))))) ;; challenge: figure out what that last one returns, before checking interpreter