1. We denote the largest common divisor of the positive integers a1, a2, . . . , an with gcd(a1, a2, . . . , an). As in the case of
   two numbers, this is the largest positive integer that divides all the ai
   :na; for example, gcd(12, 28, 48) = 4. Do any of the
   following implications apply?
   gcd(a, b, c) = 1 ⇒ gcd(a, b) = gcd(a, c) = gcd(b, c) = 1 (1)
   gcd(a, b) = gcd(a, c) = gcd(b, c) = 1 ⇒ gcd(a, b, c) = 1 (2)
   ¨Does the answer change if you know that a
   2 + b
   2 = c
   2
   ? The answer must be justified, just yes
   or no is not enough.
   1
2. Show that the set of rational numbers\ x such that x^2<2 has no minimum upper limit in Q, for example, by the
   following steps: Assume that a^2<2 and define C by a^2+c=2. Show that a<3/2. Show that if 0<\epsilon<c/4 and
   \epsilon<1, then (a+\epsilon)^2<2.
3. Section 2.1 of Stillwell describes Euclid’s algorithm. Explain how that description is related to the one in the algebra
   compendium in Mathematics I.
   Give a proof of the division algorithm: Let a and b be two integers and b >0. Then there are unambiguously
   determined numbers q and r such that a =qb+r and 0 ≤r <b. Let a and b be two quantities, for example lengths of
   distances, and carry out Euclid’s operation which Stillwell writes about, that is, replace a, b with the pair a − b, b and so
   on. Prove that this process ends if and only if a and b have a rational relationship.
4. It can be shown that the ratio between the diagonal and the side in a regular pentagon is the positive root of the
   equation x
   2 = x + 1. Use task 2 to show that the ratio (which is the so-called golden ratio) is not rational.
5. Verify your statement about the triangles in the parabola on page 10-11 (Figure 8) in Stillwell.
6. Infinity is a recurring theme in Stillwell’s book and in the context of mathematical philosophy a distinction is made
   between current and potential infinity. What kind of infinity is it in
   lim
   x→∞ x
   1
   = 0 ?
   Why?
7. Reflect on the calculations
   Z ∞ dx
   x

2


−
1
x
∞
= −

1
∞
−
1
1

= −(0 − 1) = 1
1 1
and
Z ∞
2
dx
x

2 + x − 2

1
3
Z ∞
2

1
x − 1
−
1
x + 2

dx

1
3
Z ∞
2
dx
x − 1
−
Z ∞
2
dx

x + 2

1
3
([log(x − 1)]∞
2 − [log(x + 2)]∞
2

)

1
3

(log ∞ − log 1 − (log ∞ − log 4))

1
3

(∞ − 0 − ∞ + log 4)

1
3

(∞ − ∞ + log 4)

1
3
log 4.
also
lim
x→∞
(x + 2 − x) = lim
x→∞
(x + 2) − limx→∞
x = ∞ − ∞ = 0
lim x→∞
(x + 2 − x) = lim x→∞
2 = 2.

8. Is area a property that an area has in itself or is it something we define, that is, assign it? How would Euclid
   view this? Can you relate the discussion on page 10 in Stillwell to this question?
   2
9. Let b1, b2 and b3 be the lengths of the sides in a triangle and h1, h2 and h3 respectively heights. If one
   assumes that the area of the triangle is a property of the triangle itself, then it follows that
   b1h1

2

b2h2

2

b3h3
2
.
(3)
But if the area of a triangle is something we deny, then we have to show it in some other way (??). How do
you do that? Tip: Show that the triangles 4ADC and 4BEC are uniform.

10. Determine the conditions of the numbers a, b and c so that the line ax + by + c = 0 touches the
    parabola y = x
    2
    . You can use derivatives, but I rate the solution higher if you provide a purely algebraic
    solution. See Stillwell section 4.2 for an explanation of what I mean by an algebraic solution
11. There is plenty of more or less different evidence for Pythagoras’ theorem.
    Here are two of them, which are said to originate from India.
    In the first proof, only the left figure is used. The area of the large square can be written on the one hand
    (a + b)
    2
    and on the other 4 · ab/2 + c
    2
    . So is
    (a + b)
    2 = 4 ·
    ab
    2

• c
  2
  3
  which is simplified to a
  2 + b
  2 = c
  2
  .In the second proof, both figures are used.
  The squares have the same area and contain 4 copies of the triangle.
  Thus the area of square C is equal to the area of A plus the area of B, which is Pythagoras’ theorem.
  Which axioms and geometric theorems are used more or less implicitly in the evidence? (“Implicit” means unspoken,
  that is, the sentences are used without being said.)

10. This task is about the so-called disk formula for the volume of a body in space (compare with Cavalieri’s
    principle). A body lies along the x-axis between x = a and x = b. The area of a section perpendicular to the x-axis
    at x is A(x).
    An “in nitesimally thin” disk with thickness dx has the volume A(x) dx, so the body volume is
    V =
    Z b
    a
    A(x) dx.
    Is it necessary to use infinitesimally thin slices in the proof or can you complete it without such? By the way, are ¨
    there infinitesimally small quantities?
11. Two people play a simple dice game. A round consists of throwing the dice each time and winning at most (if
    the players hit the same, they turn over). The one who has first won five rounds wins the game and the pot, ie
    the money wagered. Suppose they have to interrupt the game before someone has won and they want to split
    the pot fairly.
    What can fairness mean in this context? Solve the problem with your definition of fairness if the players cancel
    when one has won 4 times and the other 3 times.
12. The number of atoms of a radioactive substance which decomposes over a short period of time is proportional
    to the total number of atoms. If the mass of the radioactive substance at time t ¨is m(t), so therefore
    differentialekva-tionen m0
    (t) = −λm(t), where λ is a positive constant that is characteristic of the substance
    and is usually called the decay constant. Solve the equation and determine the relationship between the decay
    constant and the half-life of the substance. What is the probability that a carbon-14 atom will decay within
    1000 years if the half-life is 5730 years? Can you relate radioactive decay to any of the theories of probability?