ASUNCION, CHELZEAH P.
2021-09448
MATH 10. Mathematics, Culture, and Society
Homework 03
Part I. Let ℤ5 = {0,1,2,3,4}.
1. Construct the operation table for x5 on the set ℤ5
x5 0 1 2 3 4
0 0 0 0 0 0
1 0 1 2 3 4
2 0 2 4 1 3
3 0 3 1 4 2
4 0 4 3 2 1
2. The identity element of ℤ5 under x5 is 1 because for any a in ℤ5, a x5 1 = a. The
identity element is a special type of element in a set which leaves other elements
unchanged when combined with them. In the case of multiplication modulo 5, for any
element ‘a’ in ℤ5, when you multiply ‘a’ with the identity element, the result is ‘a’
itself. For example, 2 ×5 1 = 2, 3 ×5 1 = 3, and so on. Hence, 1 is the identity
element under ×5.
3. The inverse of each element of ℤ5 with respect to x5 is as follows:
Element inverse with respect to x5
0 DNE
1 1
2 2
3 3
4 4
Eg. 2 ×5 3 = 1, hence 3 is the inverse of 2 under ×5. Moreover, 0 does not
have an inverse under ×5 because there is no number in ℤ5 that we can multiply with
0 to get 1.
4. The solutions to the equations in ℤ5 are:
(i) For (3 ×5 x) +5 2 = 1, the solution is x = 1.
To find ‘x’ such that when we multiply 3 with ‘x’, add 2 to the
result, and then take the remainder when divided by 5, we get 1. The
solution is then x = 1 because (3 ×5 1) +5 2 = 1.
(ii) For (3 +5 x) x5 2 = 1, the solution is x = 0.
To find ‘x’ such that when we add 3 and ‘x’, multiply the result
with 2, and then take the remainder when divided by 5, we get 1. The
solution is x = 0 because (3 +5 0) ×5 2 = 1.
Part II. Table of Properties of the different geometries.
Euclidean Hyperbolic geometry Elliptic geometry
geometry
Sum of angles Have angles Have angles Have angles
in a triangle summing 180 summing to less than summing to more
degrees 180 degrees than 180 degrees
Parallel lines Have exactly Has infinitely many Have no line parallel
one line parallel lines parallel to a to a given line to a
to a given line given line through a given line through a
through a given given point given point
point
Curvature Zero Curvature Negative Curvature Positive Curvature
Area of a Circle Proportional to Greater than the Less than the square
the square of its square of its radius of its radius
radius
Circumference Proportional to Greater than 2 pi Lesser than 2 pi
of a circle its radius times its radius times its radius
Part III. Distinguish between a finite number and an infinite number.
Does INFINITY exist in the real world?
As someone who has always been intrigued by numbers, I’ve come to understand
that a finite number is a specific, countable value while an infinite number is an
immeasurable quantity, so vast that it cannot be quantified or counted. Infinity, to me,
is more of an abstract concept than a concrete number, signifying a limitless or
unending quantity in mathematics. Although I know that infinity doesn’t physically
exist in the real world, it is said that it is an invaluable concept in fields like
mathematics, physics, and philosophy. It facilitates our understanding and modeling
of scenarios where entities expand without limit or processes continue indefinitely.
For instance, we often perceive the universe’s extent as infinite, and in our calculus
studies, we’ve used infinity to compute limits and integrals (Math 27 & 28 days :>).
The concept of infinity also extends to other areas of our lives, such as in computer
science where an infinite loop is a sequence of instructions in a computer program
which loops endlessly. In the field of art, we’ve also seen the infinity symbol (∞) often
representing endless love or eternity. My experience with the Möbius strip, a surface
with only one side and one boundary curve, also provides a tangible experience of
this abstract concept. As I travel along the surface of the Möbius strip, I can keep
going indefinitely without ever reaching an edge, much like the concept of infinity.
Despite its abstract nature, infinity, a subject of centuries-long discussion among
philosophers and mathematicians, is then a fundamental part of many mathematical
theories and physical models, challenging our understanding of the world and the
mathematical rules we use to describe it.
References
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12, 2023, from https://brilliant.org/wiki/hyperbolic-geometry/
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Retrieved April 12, 2023, from
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Hyperbolic geometry | Non-Euclidean, Lobachevsky, Bolyai. (n.d.). Khan Academy.
Retrieved April 12, 2023, from
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