Making larger expressions easier to deal with

Suppose we had the fairly large expression

3 + 4 + 5 + 6 + 1 (We'll call this particular expression "Our Expression", version 1)

In a moment we're going to see how our expression can do some work for us in the real world. However, before we do that, we can use what we've learned in section 3.16 to shorten the expression, and make it easier to deal with.

First, lets evaluate some smaller parts of the large expression, so that we can replace those parts of the expression.

The expression is:

3 + 4 + 5 + 6 + 1

However, we know that 3 + 4 evaluates to 7

so we can rewrite

3 + 4 + 5 + 6 + 1 (our expression, version 1)

as

7 + 5 + 6 + 1 (our expression, version 2)

In version 2 of our expression, we've replaced 3 + 4 with 7.

Similarly, 5 + 6 evaluates to 11, so we can continue by writing

7 + 11 + 1 (our expression, version 3) (Why can we do this?)

Now, 11 + 1 evaluates to 12, so we can rewrite the above expression as

7 + 12 (our expression, version 4) (Why can we do this?)

and 7 + 12 evaluates to 19 so we can rewrite the above expression as

19 (our expression, version 5) (Why can we do this?)

This means that any time we come across version 1 of our expression, 3 + 4 + 5 + 6 + 1, we can replace it with version 5 of our expression, 19. We have found a mathematical short cut.

Here's the whole process written out

3 + 4 + 5 + 6 + 1 (our expression, version 1)
7 + 5 + 6 + 1 (our expression, version 2)
7 + 11 + 1 (our expression, version 3)
7 + 12 (our expression, version 4)
19 (our expression, version 5)

Now that we have established this, we don't have to work it out every time. If we see 3 + 4 + 5 + 6 + 1, we can just replace it with 19, because we know that 3 + 4 + 5 + 6 + 1 evaluates to 19 . We can also go one step further and say "We have now shown that it is TRUE that 3 + 4 + 5 + 6 + 1 is equal to 19". Written symbolically, we can say

It is TRUE that 3 + 4 + 5 + 6 + 1 = 19
 

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Copyright Jen Schellinck, 2006