2.17. EXERCISES 37

so, you would have −mn = b

a and so the ratio of a,b is rational after all. Even though youcannot get 0 (which you can get if the ratio of a and b is rational) you can get such aninteger combination arbitrarily small. Dirichlet did this in the 1830’s long before Dedekindconstructed the real numbers in 1858, published in 1872.

Theorem 2.16.1 If a,b are real numbers and a/b is not rational, then for everyε > 0 there exist integers m,n such that |ma+nb|< ε .

Proof: Let PN ,N ≥ 1 denote all combinations of the form ma + nb where m,n areintegers and |m| , |n| ≤ N. Thus there are (2N +1)2 of these integer combinations and allof them are contained in the interval I ≡ [−N (|a|+ |b|) ,N (|a|+ |b|)] . Now pick an integerM such that

(2N)2 < M < (2N +1)2

I know such an integer exists because (2N +1)2− (2N)2 = 4N +1 > 2. Now partition theinterval I into M equal intervals. If l is the length of one of these intervals, then

lM = 2N (|a|+ |b|) , so (2N)2 l < 2N (|a|+ |b|) and l <(|a|+ |b|)

2N≡ C

N

Now as mentioned, all of the points of PN are contained in I and there are more of thesepoints, (2N +1)2 than there are intervals, M. Therefore, some interval contains two pointsof PN .3 But each interval has length no more than C/N and so there exist k, k̂, l, l̂ integerssuch that ∣∣ka+ lb−

(k̂a+ l̂b

)∣∣≡ |ma+nb|< CN

Now let ε > 0 be given. Choose N large enough that C/N < ε . Then the above inequalityholds for some integers m,n.

2.17 Exercises1. Let z = 5+ i9. Find z−1.

2. Let z = 2+ i7 and let w = 3− i8. Find zw,z+w,z2, and w/z.

3. If z is a complex number, show there exists ω a complex number with |ω| = 1 andωz = |z| .

4. For those who know about the trigonometric functions 4, De Moivre’s theorem says[r (cos t + isin t)]n = rn (cosnt + isinnt) for n a positive integer. Prove this formulaby induction. Does this formula continue to hold for all integers n, even negativeintegers? Explain.

5. Using De Moivre’s theorem from Problem 4, derive a formula for sin(5x) and one forcos(5x). Hint: Use Problem 18 on Page 25 and if you like, you might use Pascal’striangle to construct the binomial coefficients.

3This is called the pigeon hole principle. It was used by Jacobi and Dirichlet. Later, Besicovitch used it inhis amazing covering theorem. In terms of pigeons, it says that if you have more pigeons than holes and theyeach need to go in a hole, then some hole must have more than one pigeon. In contrast to Dirichlet, Jacobi andothers who had common sense, this simple observation seems to have not been understood by people like BrighamYoung and the Utah Mormons who made polygyny (multiple wives for a single man) a religious expectation inthe 1850’s. In Utah there were more males than females.

4I will present a treatment of the trig functions which is independent of plane geometry a little later.