Page 66 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

Now, what about

1
77 + ?
2q

1 1 1 p
77 + > 77, and77+ < 77 + ≤ = x,
2q 2q q q

so

1 1 p
77 < 77 + < 77 + ≤ = x.
2q q q

Since 77 ∈ N , and q ∈ N ,

1
77 + ∈ Q.
2q

So we have found a rational number which is closer to 77 than x is and is still bigger than
77. That contradicts the definition of x .

Next, if you have two rational numbers x and y ∈ Q such that

x< y,

then you know that

0 <y − x.

What is a rational number between 0 and y − x ? What happens if you add x to this number?

• Hint for #10: Try this with 3 pennies, 2 of which are heads-up. Think about all the
possibilities, and then use your logic to find the general strategy.

Please keep in mind that you don’t always need to solve problems the way the hints suggest. In
mathematics, there are often several different ways to prove the same statement. For example, there
are many correct but different proofs of the Prime Number Theorem, which you’ll learn about in
the last chapter. So, if you are able to solve a problem in a way other than suggested in the hints and
examples, please check carefully that your steps are correct, and if so, GOOD WORK! Enjoy your

mathematical creativity!

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