1 Arithmetic and Roman Numerals

Multiplication and Division

You should feel comfortable doing addition, subtraction, multiplication, and division without a calculator.

For multiplication and division, you should know the multiplication table up to 12×12.

1	2	3	4	5	6	7	8	9	10	11	12
2	4	6	8	10	12	14	16	18	20	22	24
3	6	9	12	15	18	21	24	27	30	33	36
4	8	12	16	20	24	28	32	36	40	44	48
5	10	15	20	25	30	35	40	45	50	55	60
6	12	18	24	30	36	42	48	54	60	66	72
7	14	21	28	35	42	49	56	63	70	77	84
8	16	24	32	40	48	56	64	72	80	88	96
9	18	27	36	45	54	63	72	81	90	99	108
10	20	30	40	50	60	70	80	90	100	110	120
11	22	33	44	55	66	77	88	99	110	121	132
12	24	36	48	60	72	84	96	108	120	132	144

Practice – Multiplication

Use this to practice your multiplication table

Practice – Division

Use this to practice your multiplication table for division

You should get to the point where you know these without hesitation. A strong foundation here will set you up well for success throughout the rest of this course.

Order of Operations

When doing arithmetic over many numbers, the order in which they are completed matter!

To see why order matters, it may be easier to see through a word problem.

Example 1

John buys 4 books for $8 each and 3 pens for $2 each. How much did John spend in total?

You may write out this problem as follows:

4×8+3×2

If you solve this from left to right you will get

4×8=32+3=35×2=$70

However, this is not the correct answer!

Lets work through it as the problem reads

John buys 4 books for $8 each: 4×8=32

John buys 3 pens for $2 each: 3×2=6

Now if you add them you get 32+6=$38

The correct answer, John spent a total of $38

As you can see here the order in which you solve a problem will give you different answers.

What is the right order to solve a problem?

P E M D A S is the correct order! Lets break it down:

P – Parenthesis, this means anything in a parenthesis should go first (). If there are multiple sets of parenthesis within each other, start with the most nested.

E – Exponents, any exponents should be completed next

M/D – Multiplication and Division, do these from left to right!

A/S – Addition and subtraction, do from these left to right

Overall you should complete them from most nested, and left to right.

If you have trouble remembering this you can remember the phrase: “Please Excuse My Dear Aunt Sally”

Example 2

Compute the following

$(13-4)^2-(11\cdot4)$

First solve in parenthesis

$(9)^2-(44)$

Next, exponents

$81-44$

Finally subtraction

$\fbox{$37$}$

Roman Numerals

Many chemicals and drug names involve roman numerals, this section will go over how to read roman numerals.

Roman Numerals follow a pattern to them, which is easiest to see in the first 10 digits.

Arabic Number	Roman Numeral	Note
1	I	On the 1st-3rd numbers you will add one symbol to add
2	II
3	III
4	IV	On the 4th/9th digits you will put one mark before the next symbol to subtract
5	V
6	VI	On the 6th-8th digits you will add one of the previous symbols to add
7	VII
8	VIII
9	IX	On the 4th/9th digits you will put one mark before the next symbol to subtract
10	X

Now you may be wondering why the notes are written as “symbol” or “digit” this is because all numbers follow this similar pattern as you increase.

Here is a table of the different roman numeral symbols

Arabic Number	Roman Numeral
1	I
5	V
10	X
50	L
100	C
500	D
1000	M

As you see in this table, there are many different letters used to symbolize a number. For numbers between these symbols, the patterns listed previously will apply here.

Example 3

Write 367 in roman numerals

Start by breaking the number apart:

367 = 300 + 60 + 7

Use these to write out your roman numeral from left to right

Write 300

Since this is the third number in the set of hundreds, we will write three of the one hundred symbols: CCC

Write 60

We are looking at the sixth number in the set of tens, we write the symbol for fifty, plus one ten symbol: LX

Write 7

Finally, we look at the seventh number in the set of ones, we write the symbol for five, plus two one symbols: VII

Now we put all of these together to get the final answer

$\fbox{CCCLXVII}$

1	2	3	4	5	6	7	8	9	10	11	12
2	4	6	8	10	12	14	16	18	20	22	24
3	6	9	12	15	18	21	24	27	30	33	36
4	8	12	16	20	24	28	32	36	40	44	48
5	10	15	20	25	30	35	40	45	50	55	60
6	12	18	24	30	36	42	48	54	60	66	72
7	14	21	28	35	42	49	56	63	70	77	84
8	16	24	32	40	48	56	64	72	80	88	96
9	18	27	36	45	54	63	72	81	90	99	108
10	20	30	40	50	60	70	80	90	100	110	120
11	22	33	44	55	66	77	88	99	110	121	132
12	24	36	48	60	72	84	96	108	120	132	144

1	2	3	4	5	6	7	8	9	10	11	12
2	4	6	8	10	12	14	16	18	20	22	24
3	6	9	12	15	18	21	24	27	30	33	36
4	8	12	16	20	24	28	32	36	40	44	48
5	10	15	20	25	30	35	40	45	50	55	60
6	12	18	24	30	36	42	48	54	60	66	72
7	14	21	28	35	42	49	56	63	70	77	84
8	16	24	32	40	48	56	64	72	80	88	96
9	18	27	36	45	54	63	72	81	90	99	108
10	20	30	40	50	60	70	80	90	100	110	120
11	22	33	44	55	66	77	88	99	110	121	132
12	24	36	48	60	72	84	96	108	120	132	144

1	2	3	4	5	6	7	8	9	10	11	12
2	4	6	8	10	12	14	16	18	20	22	24
3	6	9	12	15	18	21	24	27	30	33	36
4	8	12	16	20	24	28	32	36	40	44	48
5	10	15	20	25	30	35	40	45	50	55	60
6	12	18	24	30	36	42	48	54	60	66	72
7	14	21	28	35	42	49	56	63	70	77	84
8	16	24	32	40	48	56	64	72	80	88	96
9	18	27	36	45	54	63	72	81	90	99	108
10	20	30	40	50	60	70	80	90	100	110	120
11	22	33	44	55	66	77	88	99	110	121	132
12	24	36	48	60	72	84	96	108	120	132	144