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Extra info for Blast into Math - A Fun and Rigorous Introduction to Pure Mathematics
Example text
Sets of numbers: mathematical playground 2. Next, we hold on to the handrail; we do this by assuming the statement is true for some n ∈ N which is at least as large as the base case. This is known as the induction assumption. 53 Blast into Math! Sets of numbers: mathematical playground 54 Blast into Math! Sets of numbers: mathematical playground 3. The third step is the hardest: we must get the escalator to start moving! To do this, we prove that the statement is true for n + 1 . This is like jumping up high and landing with a thump on the escalator, and BOOM!
First, what are we adding? This time, we are adding 5 times k . Where do we start? We start with k = 8 . When do we stop? We stop with k = 10 . So, 10 k=8 5k = 5 ∗ 8 + 5 ∗ 9 + 5 ∗ 10 = 40 + 45 + 50 = 135. What is 5 j2 ? j=2 Well, this time I called it j instead of k , just to emphasize that there isn’t anything special about using the letter k . What matters is the meaning. Here, 5 j2 j=2 means we sum the integers squared. Where do we start? We start with 2 . Where do we stop? We stop at 5. So, 5 j 2 = 22 + 32 + 42 + 52 = 4 + 9 + 16 + 25 = 54.
D 1 d d Since x = 1 ∗ x, by the multiplication rule for the rational numbers, x= ad da = . db bd Now we can put 1 into a different disguise so that he can put his friend y also in disguise. Since b ∈ N , b = 0 , and the multiplicative inverse of b is in Q so 1= b∗ b 1 b 1 = ∗ = . b 1 b b Since y = 1 ∗ y, by the multiplication rule for the rational numbers, y= bc bc = . bd bd So, by the addition and multiplication rules for rational numbers, x+y = ad + bc , bd x−y = ad − bc . bd Because b and d are in N , and the natural numbers are closed under multiplication, bd ∈ N.