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CHAPTER 1
INTRODUCTION
1. The Background
Numbers almost relate to every human life side. Donna and Lawrence
(1980:1) said that it is almost impossible to relate the real world without
encountering numbers in one way or another, for example : money, time,
quantities of objects, distance, etc. By studying mathematics we will know all
about numbers, one of them is about number operations. Number operations are
basic knowledge should be understood because in daily activites we always do
counting. The ability to count is also necessary to be able to master others topics
in mathematics. Therefore, these number operations are learned in elementary
school. Basically, number operations consist of 4 operations, they are addition,
subtraction, multiplication, and division.
2. The Purpose
The purpose of this paper is in order to readers know and understand
about:
a. The number operations include addition, subtraction, multiplication, and
division;
b. The properties of number operations.
c. The Bases of Number Ever Used
d. The History of Months
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CHAPTER 2
DISCUSSION
1. Number Operations
There are 4 kinds of operations in mathematics include Addition,
Subtraction, Multiplication, and Division.
a. Addition
Addition is the process of combining two or more numbers and arriving at
a single total (Berston and Fisher, 1982). Furthermore, Berston and Fisher (1982)
said that the numbers to be added are called addends and the final result of the
addition is called the total, sum or amount. The addition symbol is written by “+”
and it is read “plus” or “add” (Ashlock, et.al. 1983:68). It first appeard in 1456 in
an unpublished manuscript by the mathematician Johann Regiomontanus.
Example :
4 + 7 + 11 is read as 4 plus 7 plus 11, and the sum is 22.
Example :
1. Simplify 5 + 1
To model this expression, we first hop 5 units to the right. Then we hop 1 unit
to the right. Since we end up on 6, the answer is 6.
5 + 1 = 6
2. Simplify 5 + (-2)
To model this problem, we hop 5 units to the right. The second hop is 2 units
to the left (please see note below). Since we end up on 3, the answer is 3.
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5 + (-2) = 3
3. Simplify 5 + (-8)
To model this expression, we begin with a 5-unit hop to the right. The we hop
8 units to the left. Since 8 is bigger than 5, we cross over the 0 into the
negative side of the number line. We end up at -3, so the answer is -3.
5 + (-8) = -3
4. Simplify (-8) + 5
Since addition is commutative, (-8) + 5 should result in the same answer as
the previous example 5 + (-8). The model of (-8) + 5 means a hop of 8 units
to the left followed by a hop of 5 units to the right resulting in the same sum
as the previous example.
(-8) + 5 = -3
5. Simplify (-6) + 10
To model this expression, we begin by hopping 6 units to the left. Then we
hop 10 units to the right. Since we end up at 4, the answer is 4.
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(-6) + 10 = 4
6. Simplify: (-6) + (-2)
To model this problem, we make two hops to the left. The first hop is 6 units
to the left. Then we hop 2 units to the left. We end up at -8.
(-6) + (-2) = -8
7. Simplify: (-10) + (-5)
To model this expression, we make a 10-unit hop to the left, followed by a 5-
unit hop to the left.
b. Subtraction
Subtraction is the process of finding the difference between two amounts
(Berston and Fisher, 1982). There are three parts in subtraction, they are minuend,
subtrahend and difference. Minuend is a number from which another is subtracted.
It is the first number in subtraction. Subtrahend is the amount that is subtracted
from another (minuend) while the result of subtraction is called the remainder or
difference. It first appeard in 1456 in an unpublished manuscript by the
mathematician Johann Regiomontanus. Ashlock, et. al. (1983:72) said that
subtraction is the inverse of addition. Addition is used to find the sum when its
addends are known, while subtraction is used to find one of the addends when the
sum and its other addend are known. The subtraction symbol is written by “-” and
it is read “minus” or “subtract”.
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-4
-(+ 7)
+ 3
+ 10
-(+7) 3
Example :
9 – 5 is read 9 minus 5, and the difference is 4.
Examples :
1. 10 - 7 =
When we subtract a positive number, we move to the left on the number line
2. 3 – 7 =
When we subtract a positive number, we move to the left on the number line
3. I owe my brother Rp 5 thousand. I fold his pen, and he forgives Rp 3
thousand of my debt
I still owe my brother Rp 2 thousand
- 2
- (-3)
- 5
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3 – 7 = ? -4
10 – 7 = ? 3
– 5 – (-3) = ? - 2
take away debt
If I subtract a positive number, I
move left. So to subtract a
negative number, I move right
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4. A diver dove 15 feet below sea level.
Then she swam down another 10 feet.
Where is she now?
The diver is 25 feet below sea level.�
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c. Multiplication
Multiplication is form of repeated addition (Berston and Fisher, 1982).
The number that is multiplied is called multiplicand and the number that
multiplied by is called multiplier. The result of multiplication is called the product
(Berston and Fisher, 1982). The multiplication symbol is written by “x” or “�” or
“�” and it is read “multiply” or “times”. Pickover (2005:13) said that the
multiplication symbol “x” was introduced by The English mathematician William
Oughtred (1574-1660) in 1631 in his book Keys to Mathematics, published in
London.
Example :
4 x 6 is read 4 times 6, the product is 24.
Example :
1. 3 x 2 = 6
When multiplying integers, the first number tells how many sets there are.
The second number tells how many are in each set.
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–1 5 – 10 = ? - 25 �
- 15
- (+10)
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2
2
2
6
2 2
6
2
3 x 2
When the first number is positive, multiplication can be thought of as repeated
addition. 3 x 2 means add three sets of two positive number. Adding three sets of
two positive number is six positive number in total. So three times two equals six.
2. 3 x (-2) = -6
3 x (-2) can be thought of as repeated addition, add three sets of two negative
number. Adding three sets of two negative number is six negative number in
total. So three times negative two equals negative six.
Two in each sets Three sets
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d. Division
Berston and Fisher (1982) said that division is a process of finding out
how many times one number is contained in another number. Furthermore, the
dividend or numerator is the number being divided; the number by which the
dividend is being divided is the divisor or denominator; the quotient is the result
of the division and shows the number of times the divisior goes into the dividend:
and the remainder is the amount remaining if the divisor does not go into the
dividend an even number of times (Berston and Fisher, 1982). The division
symbol is written by “÷” or “/” and it is read “divided by”. Division symbol “÷”
first appeared in print in Johann Heinrich Rahn’s Teutche Alegbra in 1659
(Pickover, 2005:20). Ashlock (1983:175) said that division is the inverse of
multiplication. Multiplication is used to find a product when its factors are known,
whereas division is used to find one of the factors when the product and its other
factor are known.
Example :
16 : 2 = 8 Result
Example :
1. 6 : 3 =
The result is 6 : 3 = 2
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2. – 6 : 3 =
The result is - 6 : 3 = - 2
3. 6 : (- 3) =
The result is 6 : (- 3) = - 2
4. – 6 : ( - 3) =
The result is - 6 : (- 3) = 2
2. The Properties of Number Operations
The following are the properties (also called prinsiples or laws) of number
operations which was written by Gibilisco (2004:20-21):
Additive Identity Element
When 0 is added to any real number a, the sum is always equal to a. The
number 0 is said to be the additive identity element:
a + 0 = a
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Additive Inverses
For every real number a, there exists a unique real number -a such that the
sum of the two is equal to 0. The numbers a and -a are called additive inverses:
a + (- a) = 0
Commutative Law For Addition
When any two real numbers are added together, it does not matter in which
order the sum is performed. The operation of addition is said to be commutative
over the set of real numbers. For all real numbers a and b, the following equation
is valid:
a + b = b + a
Associative Law For Addition
When adding any three real numbers, it does not matter how the addends are
grouped. The operation of addition is associative over the set of real numbers. For
all real numbers a1, a2, and a3, the following equation holds true:
(a1 + a2) + a3 = a1 + (a2+ a3)
Commutative Law For Multiplication
When any two real numbers are multiplied by each other, it does not matter in
which order the product is performed. The operation of multiplication, like
addition, is commutative over the set of real numbers. For all real numbers a and
b, the following equation is always true:
a x b = b x a
it also can be written without “times symbol” as:
ab = ba
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Associative Law For Multiplication
When multiplying any three real numbers, it does not matter how the
multiplicands are grouped. Multiplication, like addition, is associative over the set
of real numbers. For all real numbers a, b, and c, the following equation holds:
(ab)c = a(bc)
Multiplication Identity Element
When any real number a is multiplied by 1, the product is always equal to a. The
number 1 is said to be the multiplicative identity element:
a x 1 = a
Multiplication Inverses
For every nonzero real number a, there exists a unique real number �
� such that the
product of the two is equal to 1. The numbers a and �
� are called multiplicative
inverses:
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Furthermore, Gibilisco (2004:20) said “the multiplicative inverse of a real number
is also called its reciprocal”.
Distributive Law
For all real numbers a, b, and c, the following equation holds. The operation of
multiplication is distributive with respect to addition:
a(b + c) = ab + ac
The multiplication is also distributive with respect to subtraction:
a(b - c) = ab – ac
The distributive law can also be extended to division as long as there aren’t any
denominators that end up being equal to zero. For all real numbers a, b, and c,
where a � 0, the following equations are valid:
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�!
�"
�#
�� ! " #
�! % �#$
��
�!
�%
�#
�� ! % #
The following is the order of operations:
1. Perform any calculations inside parentheses
2. Perform all multiplications and divisions, from left to right
3. Perform all addition and subtraction, from left to right
Examples:
1. 2 + 3 x 7 = 2 + 21 = 23
Reason:
2 + 3 x 7 means 2 + (7 + 7 + 7) = 2 + 21 = 23 (correct)
But, if we perform addition firstly:
2 + 3 x 7 = 5 x 7 = 35 (False)
2. 4 + 8 x (2 + 7) ÷ 12 – 7 = 4 + 8 x 9 ÷ 12 – 7
= 4 + 72 ÷ 12 – 7
= 4 + 6 – 7
= 10 – 7
= 3
3. The Bases of Number Ever Used
a. Roman Numeral System
The ancient Romans used letters of their alphabet to represent numbers,
and this method is still used in certain countries even today (Berston and
Fisher,1982). Originally the Roman numeral for five was a sketch of a
hand. Later this was simplified to a V. Observed that there is some use of
place value in Roman numerals IV and IX. The Roman system was based
on 10, since all their symbols for large numbers denote multiples of 10,
that is L = 5 x 10, C = 10 x10, etc. It is good experience to work some
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arithmetic problems using these Roman numerals. Addition and
subtraction are relatively easy, but multiplication and division are not.
b. Babylonian Numeral System
Babylonian numeral system were written using a sexagesimal (base-
60) numeral system. From this derives the modern day usage of 60
seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a
circle, as well as the use of seconds and minutes of arc to denote fractions
of a degree. Babylonian advances in mathematics were facilitated by the
fact that 60 has many divisors. Also, unlike the Egyptians, Greeks, and
Romans, the Babylonians had a true place-value system, where digits
written in the left column represented larger values, much as in
the decimal system. They lacked, however, an equivalent of the decimal
point, and so the place value of a symbol often had to be inferred from
the context. On the other hand, this "defect" is equivalent to the modern-
day usage of floating point arithmetic; moreover, the use of base 60
means that any reciprocal of an integer which is a multiple of divisors of
60 necessarily has a finite expansion to the base 60. (In decimal
arithmetic, only reciprocals of multiples of 2 and 5 have finite decimal
expansions.) Accordingly, there is a strong argument that arithmetic Old
Babylonian style is considerably more sophisticated than that of current
usage.
c. Greece Numeral System
Greece numeral system was based on 10 but had neither place value nor a
symbol for zero. They used the letters of their alphabet for numbers, but
you need not know the Greek alphabet in order to understand the system.
The numerals and some of their names were as followers.
d. Hindu-Arabic Numeral System
The number system most universally used today is known as the decimal
system. It probably originated in India and was brought to Europe by the
Arabs. The digits 1, 2, 3, ... and 0 are called Hindu-Arabic numerals. It is
a system based upon tens, and the starting point is indicated by a dot
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known as the decimal point. Whole numbers are written to the left of the
decimal point and increase by tens depending on the position of the
numerals. Numbers to the right of the decimal point indicate amounts
less than the number 1 and are also in groups of ten.
4. The History of Months
The history of months and their names begun in Rome. The original Roman
year had 10 named months Martius "March", Aprilis "April", Maius "May",
Junius "June", Quintilis "July", Sextilis "August", September "September",
October "October", November "November", December "December", and
probably two unnamed months in the dead of winter when not much
happened in agriculture. The year began with Martius "March". Numa
Pompilius, the second king of Rome circa 700 BC, added the two months
Januarius "January" and Februarius "February". He also moved the beginning
of the year from Marius to Januarius and changed the number of days in
several months to be odd, a lucky number. After Februarius there was
occasionally an additional month of Intercalaris "intercalendar". This is the
origin of the leap-year day being in February. In 46 BC, Julius Caesar
reformed the Roman calendar (hence the Julian calendar) changing the
number of days in many months and removing Intercalaris.
a. Calendar of Romulus
Roman writers attributed the original Roman calendar to Romulus, the
mythical founder of Rome around 753 BC. The Romulus calendar had
ten months with the spring equinox in the first month:
Calendar of Romulus
Martius (31 days)
Aprilis (30 days)
Maius (31 days)
Iunius (30 days)
Quintilis [2]
(31 days)
Sextilis (30 days)
September (30 days)
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October (31 days)
November (30 days)
December (30 days)
The regular calendar year consisted of 304 days, with the winter days
after the end of December and before the beginning of the following
March not being assigned to any month.
The names of the first four months were named in honour of Roman
gods: Martius in honour of Mars; Aprilis in honour of Fortuna Virilis
(later Venus Verticordia in the mid-4th century AD); Maius in honour of
Maia; and Iunius in honour of Juno. The names of the months from the
fifth month on were based on their position in the calendar: Quintilis
comes from Latin quinque meaning five; Sextilis from sex meaning six;
September from septem meaning seven; October from octo meaning
eight; November from novem meaning nine; and December from decem
meaning ten.
b. Calendar of Numa
Numa Pompilius, the second of the seven traditional kings of Rome,
reformed the calendar of Romulus around 713 BC. The Romans
considered odd number to be lucky, so Numa took one day from each of
the six months with 30 days, reducing the number of days in the 10
previously defined months by a total of six days.
There were 51 previously unallocated winter days, to which were added
the six days from the reductions in the days in the months, making a total
of 57 days. These he made into two months, January and February, which
he prefixed to the previous 10 months. January was given 29 days, while
February had the unlucky number of 28 days, suitable for the month of
purification. This made a regular year (of 12 lunar months) 355 days long
in place of the previous 304 days of the Romulus calendar. Of the 11
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months with an odd number of days, four had 31 days each and seven
had 29 days each:
Civil calendar Religious calendar
According to
Macrobius
and Plutarch
According to Ovid
(modern order due to
Decemviri, 450 BC)
According to Fowler
Ianuarius (29) Ianuarius Martius
Februarius (28) Martius Aprilis
Martius (31) Aprilis Maius
Aprilis (29) Maius Iunius
Maius (31) Iunius Quintilis
Iunius (29) Quintilis Sextilis
Quintilis (31) Sextilis September
Sextilis (29) September October
September (29) October November
October (31) November December
November (29) December Ianuarius
December (29) Februarius Februarius
February consisted of two parts, each with an odd number of days. The
first part ended with the Terminalia on the 23rd, which was considered
the end of the religious year, and the five remaining days formed the
second part.
c. Julian Calender
The Julian calendar, introduced by Julius Caesar in 46 BC (708 AUC),
was a reform of the Roman calendar. It took effect in 45 BC (709 AUC),
shortly after the Roman conquest of Egypt. It was the predominant
calendar in the Roman world, most of Europe, and in European
settlements in the Americas and elsewhere, until it was refined and
superseded by the Gregorian calendar. The difference in the average
length of the year between Julian (365.25 days) and Gregorian (365.2425
days) is 0.002%.
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The Julian calendar has a regular year of 365 days divided into 12
months, as listed in Table of months. A leap day is added to February
every four years. The Julian year is, therefore, on average 365.25 days
long. It was intended to approximate the tropical (solar) year. Although
Greek astronomers had known, at least since Hipparchus, a century
before the Julian reform, that the tropical year was a few minutes shorter
than 365.25 days, the calendar did not compensate for this difference. As
a result, the calendar year gained about three days every four centuries
compared to observed equinox times and the seasons. This discrepancy
was corrected by the Gregorian reform of 1582. The Gregorian calendar
has the same months and month lengths as the Julian calendar, but inserts
leap days according to a different rule. Consequently, the Julian calendar
is currently 13 days behind the Gregorian calendar; for instance, 1
January in the Julian calendar is 14 January in the Gregorian. Old Style
(O.S.) and New Style (N.S.) are sometimes used with dates to indicate
either whether the start of the Julian year has been adjusted to start on 1
January (N.S.) even though documents written at the time use a different
start of year (O.S.), or whether a date conforms to the Julian calendar
(O.S.) rather than the Gregorian (N.S.). Dual dating uses two consecutive
years because of differences in the starting date of the year, or includes
both the Julian and Gregorian dates.
The Julian calendar has been replaced as the civil calendar by the
Gregorian calendar in almost all countries which formerly used it,
although it continued to be the civil calendar of some countries into the
20th century. Most Christian denominations in the West and areas
evangelized by Western churches have also replaced the Julian calendar
with the Gregorian as the basis for their liturgical calendars. However,
most branches of the Eastern Orthodox Church still use the Julian
calendar for calculating the dates of moveable feasts, including Easter
(Pascha). Some Orthodox churches have adopted the Revised Julian
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calendar for the observance of fixed feasts, while other Orthodox
churches retain the Julian calendar for all purposes. The Julian calendar
is still used by the Berber people of North Africa, and on Mount Athos.
In the form of the Alexandrian calendar, it is the basis for the Ethiopian
calendar, which is the civil calendar of Ethiopia.
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CHAPTER 3
CONCLUSION AND SUGGESTION
1. Conclusion
a. There are 4 (four) number operations: addition, subtraction, multiplication,
and division. Addition is the process of combining two or more numbers and
arriving at a single total. Subtraction is the process of finding the difference
between two amounts. Division is the process of splitting into equal parts.
Multiplication is form of repeated addition.
b. The properties of number operations consist of idendity element, inverses,
commutative, assosiative, and distributive laws.
c. The Bases Number Ever Used are Roman Numeral System, Babylonian
Numeral System, Greece Numearl System, Hindu-Arabic Numeral System.
d. The History of Months and their names begun in Rome. The original Roman
year had 10 named months Martius "March", Aprilis "April", Maius "May",
Junius "June", Quintilis "July", Sextilis "August", September "September",
October "October", November "November", December "December". Numa
Pompilius, the second king of Rome circa 700 BC, added the two months
Januarius "January" and Februarius "February"
2. Suggestion
We suggest to all readers in order to master these number operations
because so necessary to be understood as basic knowledge. Especially for the
teachers who teach mathematics in english, we hope this paper become one of
references which is used.
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REFERENCES
Brumfiel,Charles F., Dkk. 1962. Fundamental Concepts of Element Mathematics.
USA : Addison-Wesley Publising Copany,INC
Pickover, Clifford A. (2005). A Passion for Mathematics : Numbers, Puzzles,
Madnes, Religion, and The Quest for Reality. USA: John Wiley & Sons,
Inc., Hoboken, New Jersey.
Gibilisco, Stan. (2004). Everyday Math. USA: The McGraw-Hill Companies, Inc.
Berston & Fisher. (1982). Collegiate Business Mathematics. Collegiate Business
Mathematics.
Ashlock, et. al. (1983). Guiding Each Child’s Learning of Mathematics: A
Diagnostic Approach to Instruction. USA: Bell & Howell Company.
Donna & Lawrence. (1980). Mathematics for Consumers with Business
Applications. Philippines: Addison-Wesley Publishing Company, Inc.
http://en.wikipedia.org/wiki/Arithmetic.htm. (online, diakses tanggal 10
September 2014)
http://en.wikipedia.org/wiki/History_of_mathematics (online, diakses tanggal 25
September 2014)
http://en.wikipedia.org/wiki/Julian_calendar (online, diakses tanggal 25
September 2014)
http://en.wikipedia.org/wiki/Roman_calendar (online, diakses tanggal 25
September 2014)
