Susanna S.Epp
Section 4.6 Problem #12
If a and b are rational numbers, b≠0, and r is
an irrational number, then a+br is irrational.
Let us first rewrite this problem in a more
mathematical format.
⋁a,b Є Q, If r is irrational and b≠0, then a+br is
irrational

Let us begin the proof
Let the negative be true
a,b Є Q, r is rational, b≠0 and a+br is rational
a+br = x/y by def of rational numbers
Let a=m/n and b=f/g since a,b Є Q
m/n + (f/g)r = x/y
(f/g)r = x/y – m/n
(f/g)r = (xn – mg)/(yn)
r = (gxn – gmy)/(fyn)
r Є Q since it can be written the form a/b
Conclusion
This contradicts that r is irrational.
Hence the original statement is true.
Proof by contradiction.
18. The diagram below shows the keypad fora. How many different PINs
are represented by the
an automatic
same sequence of keys as
teller machine. As you can see, the same
2133?
sequence of
keys represents a variety of different PINs.
b. How many different PINs
For instance,
are represented
by the
2133, AZDE, and BQ3F are all keyed in
same sequence of keys as
exactly the
5031?
same way.
c. At an automatic teller
machine, each PIN
corresponds to a four-digit
numeric sequence. For
instance, TWJM corresponds
to 8956. How many such
numeric sequences
contain no repeated digit
a. Let step 1 be to choose either the number 2 or one of the letters
corresponding to the number 2 on the keypad, let step 2 be to choose
either the number 1 or one of the letters corresponding to the number
1 on the keypad, and let steps 3 and 4 be to choose either the
number 3 or one of the letters corresponding to the number 3 on the
keypad. There are 4 ways to perform step 1, 3 ways to perform step
2, and 4 ways to perform each of steps 3 and 4. So by the
multiplication rule, there are 4*3*4*4 = 192 ways to perform the entire
operation.
b. Constructing a PIN that is obtainable by the same keystroke sequence
as 5031 can be thought of as the following four-step process. Step 1 is to
choose either the digit 5 or one of the three letters on the same key as
the digit 5, step 2 is to choose the digit 0, step 3 is to choose the digit 3
or one of the three letters on the same key as the digit 3, and step 4 is to
choose either the digit 1 or one of the two letters on the same key as the
digit 1. There are four ways to perform steps 1 and 3, one way to perform
step 2, and three ways to perform step 4. So by the multiplication rule
there are 4 *1 * 4 * 3 = 48 different PINs that are keyed the same as
5031.
c. Constructing a numeric PIN with no repeated
digit can be thought of as
the following fourstep process. Steps 1-4 are to choose the digits in
position 1-4 (counting from the left). Because no
digit may be repeated, there are 10 ways to
perform step one, 9 ways to perform step two, 8
ways to perform step three, and 7 ways to perform
step four. Thus the number of numeric PINs with no
repeated digit is 10 * 9 * 8* 7 = 5040.
Blaise Pascal
Born
June 19, 1623
Clermont-Ferrand,
France
Died
August 19, 1662
Paris, France
“Pascal, Blaise (1623-62), French philosopher, mathematician, and physicist,
considered one of the great minds in Western intellectual history. Pascal was born in
Clermont-Ferrand on June 19, 1623, and his family settled in Paris in 1629. Under the
tutelage of his father, Pascal soon proved himself a mathematical prodigy, and at the
age of 16 he formulated one of the basic theorems of projective geometry, known as
Pascal's theorem and described in his Essai pour les coniques (Essay on Conics,
1639). In 1642 he invented the first mechanical adding machine. Pascal proved by
experimentation in 1648 that the level of the mercury column in a barometer is
determined by an increase or decrease in the surrounding atmospheric pressure rather
than by a vacuum, as previously believed. This verified the hypothesis of the Italian
physicist Evangelista Torricelli concerning the effect of atmospheric pressure on the
equilibrium of liquids. Six years later, with the French mathematician Pierre de Fermat,
Pascal formulated the mathematical theory of probability, which has become important
in such fields as actuarial, mathematical, and social statistics and as a fundamental
element in the calculations of modern theoretical physics. Pascal's other important
scientific contributions include the derivation of Pascal's law or principle, which states
that fluids transmit pressures equally in all directions, and his investigations in the
geometry of infinitesimals. His methodology reflected his emphasis on empirical
experimentation as opposed to analytical, a priori methods, and he believed that
human progress is perpetuated by the accumulation of scientific discoveries resulting
from such experimentation. “
Source: http://library.thinkquest.org/10170/voca/pascal.htm#
.
A computer programming team has 13 members.
a.)How many ways can a group of seven be chosen to work on a project?
(13 C 7)
b.) Suppose seven team members are women and six are men.
(i) How many groups of seven can be chosen that
contain four women and three men?
(7 C 4)*(6 C 3)
(ii) How many groups of seven can be chosen that
contain at least one man?
(13 C 7) – (7 C 7)
(iii) How many groups of seven can be chosen that
contain at most three women?
(7 C 3)*(6 C 4) + (7 C 2)*(6 C 5) + (7 C 1)*(6 C 6)
c.) Suppose two team members refuse to work together on
projects. How many groups of seven can be chosen to
work on a project?
(11 C 6) + (11 C 6) + (11 C 7)
d.) Suppose two team members insist on either working
together or not at all on projects. How many groups of
seven can be chosen to work on a project?
(11 C 7) + (11 C 5)