1.1 Number Systems

Mathematics is often explained in school as the language of numbers , and while true, this description offers only a limited insight as to what mathematics really is. Mathematics, as we shall see, covers a broad area of subjects, including counting, measuring distances, and describing shapes .

One of the first historical uses of math was for counting; to determine how much of something existed, e.g., there are five people, or 10 apples, etc. Later, this system was extended to measure distances so people could understand spatial relationships. However, to express the concept of quantity, a system of numerals , or numbers, was required. The Romans used Roman numerals, the Greeks used letters and other symbols, but none of these systems survived antiquity completely. The numerals we use today find their origins in the Arabic numerals, such as 1, 2, 3, … etc. These standard numerals as we know them today take two forms: cardinal and ordinal .

The cardinal numbers are used to answers questions such as "How many?" and are:

1, 2, 3, 4, 5, 6, 7, and so on.

The ordinal numbers are used to answer questions such as "Which one?" and are:

1st, 2nd, 3rd, 4th, 5th, 6th, 7th, and so on.

1.1.1 Decimal System (Base 10)

The standard numerals, also known as the counting numbers, are arranged in a system of notation. The system we use now is known as the decimal system , and every number in the decimal system can be composed from the following 10 numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Whenever we write a number in decimal form we do so using a notation of place value . For example, the number 351 can be written in decimal as follows .

Hundreds	Tens	Units
3	5	1

Each column represents the next power of 10 from the one on the right (1 × 10, 10 × 10, 100 × 10), and this means smaller quantities are written on the right and larger ones on the left. 351 can be seen to represent 3 hundreds, 2 tens, and 1 unit.

Beyond hundreds, tens, and units, there are thousands, millions, billions, trillions, and so on. When a decimal number is greater than three digits, such as 4510 or 7900, it is typical to see a comma inserted every three digits from the right to break the number down and make it easier to read, such as 4,510 and 7,900. The comma system is extended to every three digits, as in 973,452,576,999.

Note

Each column, whether hundreds, tens, units, or whatever, is called a number's place value . If a number has 3 hundreds, no tens, and 2 units, it cannot be written as 32. It should appear as 302. The 0 is inserted into any column to indicate that there is no value.

1.2 Arithmetic

Nowadays, people often use the word "arithmetic" to mean mathematics in general. However, arithmetic actually refers to the basic operations that occur to numbers , such as addition, subtraction, multiplication, and division. Using these operations together can form expressions in which various values can be represented. These operations will now be considered .

1.2.1 Addition

If I am the owner of one tomato, and after searching through the garden I find two more tomatoes, I will have increased my total of tomatoes to three. This process of combining two or more quantities is called addition , and the result of an addition is called the sum . So, the sum of 1 + 2 is 3.

1.2.2 Subtraction

If after collecting three tomatoes, someone comes along and steals one of my tomatoes, I shall be left with two. The process of reducing one quantity by another is called subtraction , and the result of subtraction is called the difference . So, the difference of 3 − 1 is 2. Subtraction is the opposite (or inverse ) of addition.

1.2.3 Multiplication

Increasing a quantity in terms of multiples means to multiply a quantity. For example, 2 × 3 is the same as 3 + 3 + 3. And 4 × 5 is the same as 5 + 5 + 5 + 5. The result of multiplication is called the product . So, the product of 7 × 3 is 21.

1.2.4 Division

Division is the opposite (or inverse ) of multiplication. Rather than increasing a quantity, division is the process of determining how many times one quantity fits into another. The result of division is called the quotient . The quantity to be divided is called the dividend , and the quantity it's divided by is called the divisor . So the quotient of 8 ÷ 2 is 4, because 2 (the divisor) fits into 8 (the dividend) exactly four times.