It has 4 properties

1)commutative
2)associative
3)identity/zero
4)distributive

Examples of Subtraction

* The following are simple subtraction problems.
5 – 5 = 0
2 – 4 = – 2
6 – 5 = 1
10 – 6 = 4
5.3 – 1.1 = 4.2

Properties of Addition

There are four properties involving addition that will help make problems easier to solve. They are the commutative, associative, additive identity and distributive properties.

Commutative property: When two numbers are added, the sum is the same regardless of the order of the addends. For example 4 + 2 = 2 + 4

Associative Property: When three or more numbers are added, the sum is the same regardless of the order of addition. For example (2 + 3) + 4 = 2 + (3 + 4)

Additive Identity Property: The sum of any number and zero is the original number. For example 5 + 0 = 5.

Distributive property: The sum of two numbers times a third number is equal to the sum of each addend times the third number. For example 4 * (6 + 3) = 4*6 + 4*3

It seems that each student interpreted the problem differently, resulting in two different answers. Student 1 performed the operation of addition first, then multiplication; whereas student 2 performed multiplication first, then addition. When performing arithmetic operations there can be only one correct answer. We need a set of rules in order to avoid this kind of confusion. Mathematicians have devised a standard order of operations for calculations involving more than one arithmetic operation.

Rule 1: First perform any calculations inside parentheses.
Rule 2: Next perform all multiplications and divisions, working from left to right.
Rule 3: Lastly, perform all additions and subtractions, working from left to right.

The above problem was solved correctly by Student 2 since she followed Rules 2 and 3. Let’s look at some examples of solving arithmetic expressions using these rules.
Example 1: Evaluate each expression using the rules for order of operations.
Solution:
Order of Operations
Expression Evaluation Operation
6 + 7 x 8 = 6 + 7 x 8
Multiplication
= 6 + 56 Addition
= 62
16 ÷ 8 – 2 = 16 ÷ 8 – 2 Division
= 2 – 2 Subtraction
= 0
(25 – 11) x 3 = (25 – 11) x 3 Parentheses
= 14 x 3 Multiplication
= 42

In Example 1, each problem involved only 2 operations. Let’s look at some examples that involve more than two operations.
Example 2: Evaluate 3 + 6 x (5 + 4) ÷ 3 – 7 using the order of operations.
Solution:
Step 1: 3 + 6 x (5 + 4) ÷ 3 – 7 = 3 + 6 x 9 ÷ 3 – 7 Parentheses
Step 2: 3 + 6 x 9 ÷ 3 – 7 = 3 + 54 ÷ 3 – 7 Multiplication
Step 3: 3 + 54 ÷ 3 – 7 = 3 + 18 – 7 Division
Step 4: 3 + 18 – 7 = 21 – 7 Addition
Step 5: 21 – 7 = 14 Subtraction
Example 3: Evaluate 9 – 5 ÷ (8 – 3) x 2 + 6 using the order of operations.
Solution:
Step 1: 9 – 5 ÷ (8 – 3) x 2 + 6 = 9 – 5 ÷ 5 x 2 + 6 Parentheses
Step 2: 9 – 5 ÷ 5 x 2 + 6 = 9 – 1 x 2 + 6 Division
Step 3: 9 – 1 x 2 + 6 = 9 – 2 + 6 Multiplication
Step 4: 9 – 2 + 6 = 7 + 6 Subtraction
Step 5: 7 + 6 = 13 Addition

In Examples 2 and 3, you will notice that multiplication and division were evaluated from left to right according to Rule 2. Similarly, addition and subtraction were evaluated from left to right, according to Rule 3.

When two or more operations occur inside a set of parentheses, these operations should be evaluated according to Rules 2 and 3. This is done in Example 4 below.
Example 4: Evaluate 150 ÷ (6 + 3 x – 5 using the order of operations.
Solution:
Step 1: 150 ÷ (6 + 3 x – 5 = 150 ÷ (6 + 24) – 5 Multiplication inside Parentheses
Step 2: 150 ÷ (6 + 24) – 5 = 150 ÷ 30 – 5 Addition inside Parentheses
Step 3: 150 ÷ 30 – 5 = 5 – 5 Division
Step 4: 5 – 5 = 0 Subtraction

Example 5: Evaluate the arithmetic expression below:

Solution: This problem includes a fraction bar (also called a vinculum), which means we must divide the numerator by the denominator. However, we must first perform all calculations above and below the fraction bar BEFORE dividing.

Thus
Evaluating this expression, we get:

Example 6: Write an arithmetic expression for this problem. Then evaluate the expression using the order of operations.
Mr. Smith charged Jill $32 for parts and $15 per hour for labor to repair her bicycle. If he spent 3 hours repairing her bike, how much does Jill owe him?
Solution: 32 + 3 x 15 = 32 + 3 x 15 = 32 + 45 = 77
Jill owes Mr. Smith $77.
Summary: When evaluating arithmetic expressions, the order of operations is:

* Simplify all operations inside parentheses.
* Perform all multiplications and divisions, working from left to right.
* Perform all additions and subtractions, working from left to right.

If a problem includes a fraction bar, perform all calculations above and below the fraction bar before dividing the numerator by the denominator.

* Numerals are written with the largest values to the left: MCI is one thousand plus one hundred plus one, or 1101.
* The values of two identical adjacent numerals are added. MMCI is one thousand plus one thousand plus one hundred plus one.
* A numeral that is “out of order,” that is, that appears to the left of a numeral with a larger value, has its value subtracted from the value of the larger numeral. So for example:

IV is 4 but VI is 6
IX is 9 but XI is 11
MCM is 1900 but MMC is 2100

* The switch to using subtraction generally is made at the point where four adjacent identical numerals would be needed if only addition were used. So:

4 = IV not IIII
9 = IX not VIIII
40 = XL not XXXX
90 = XC not LXXXX
400 = CD not CCCC
900 = CM not DCCCC

* Roman numerals are very decimally oriented. Reading from left to right, a Roman numeral consists of distinct subgroups, each of which represents what would be a place in a Hindu-Arabic number. Only a particular set of numerals are proper in each subgroup: those that describe values in that range.
Roman numerals encountered today usually begin with, at most, M’s and only M’s. Counting them gives the value of the thousands place in a decimal number. Place the mouse cursor over the Roman numeral below to split it into its decimal groups. Place the mouse cursor over the yellow areas to make the numbers count up, and see how they change.

1947
MCMXLVII
1 9 4 7
M CM XL VII

The hundreds group can include only C’s; M minus C (CM); D; and D minus C (CD).

The tens group can include only X’s; C minus X (XC); L; and L minus X (XL)

The units group can include only I’s; X minus I (IX); V; and V minus I (IV).
49 = XLIX not IL
99 = XCIX not IC
1999 = MCMXCIX not MIM
990 = CMXC not XM

The effect is that only I’s, X’s and C’s are subtracted, and only from, at most, the next two larger numerals. This convention greatly eases the reader’s burden, by substituting recognition for calculation. For example, anyone who has been reading copyright dates on many books immediately recognizes a date beginning “MCM…” as something from the 1900′s. If “MIM” were permitted, the reader would have to actually do the subtraction.

* In the past Roman numerals included symbols other than those in the above table, some until quite recently. As late as the 1960′s the United States Government Printing Office used the convention that a bar over a letter multiplied its value by 1000, so V with a bar is 5000, and M with a bar is 1,000,000. This usage is now very rare.
* Today only addition and subtraction are employed in reading a number written in Roman numerals. But see below.

Today, Roman numerals are used mainly as an alternative to the Hindu-Arabic numerals in outlines and other instances in which two distinct sets of numerals are useful, for clock faces, for ceremonial and monumental purposes, and by publishers and film distributors who have an interest in making copyright dates difficult to read.
Roman numerals in ancient Rome

Perhaps the biggest difference between modern and Roman Roman numerals is that the Romans rarely used the subtraction principle. Nine was much more likely to be VIIII than IX.

A line drawn over a numeral meant that its value was to be multiplied by 1000. If lines were drawn on the top and both sides of a numeral, its value was multiplied by a hundred thousand.

(|)
Roman numerals in Europe during the Middle Ages

This system was almost the only one used in Europe until about the 11th century, and was gradually supplanted during the next 500 years by Hindu-Arabic numerals.

In the Middle Ages, a few conventions no longer used were common.

To make altering the last digit in a number more difficult, the final “i” was extended. Until recently, this practice was often represented in print by j instead of i, for example viij instead of viii.
If a number was followed by a raised, smaller number (as we would write an exponent), multiply the two numbers.

A very common use of this technique was to indicate a number of scores. For example:

“For there is a C of vixx thereby be sold muttons and other beasts and fishes, as for herring vxx with the tale herring make a C: xM make a last; and because that a MI wyll not in a barrel, therefore xii barrels packed herring make a last.”

–MS Cotton, Vesp. E. IX (15th century)

vixx = 6 times 20, i.e., 6 score, = 120; muttons were sold by a “hundred” (“C”) of 120 pieces

vxx = 5 times 20, i.e., 5 score, = 100; herring was sold by a hundred of 100 pieces

xM = 10 times 1000 = 10,000

MI = 1000
The Roman numerals C and M sometimes did not mean 100 or 1000 (see hundred).

A phrase was frequently added to resolve the ambiguity. For example, from the same source as above:

“Also eels be sold by the stike, that is xxv eels, and x stikes make a gwyde, iicl by vxx.” (ibid)

Here the phrase “by vxx” (five score) indicates that the “c” in the previous number means 100, so iicl is 250 (2 times 100, plus 50).
Roman numerals in old manuscripts

Certain types of errors are typical in reading Roman numerals in old manuscripts, due to physical damage to the text. Kemble describes some:

This [inconsistency in dates] however generally arises from the latter date having been partially abraded by age, and so misread: the want of a light line at the bottom readily transforms a V (in the old charters U) into a II; an abrasion may convert an X into a V: hence we not uncommonly find in these copies indiction IIII for VII or XV; VII for XII; XII for XV, and the like. Nor is another error at all uncommon, where a letter or contraction has been taken to be part of the date: for instance, indictione uo (uero) IIa, has often been read as if it were indictione VIIa. Again, indictione Xma has become transformed into indictione XIIIa, the strokes of the written “m” having been taken to represent three units. This cause of error is so frequent as to render multiplied examples unnecessary.

Symbol Meaning
I 1
V 5
X 10
L 50
C 100
D 500
M 1,000