**Vocabulary**

There are four basic operations in mathematics: addition, subtraction, multiplication, and division. Often when we talk about a collection of numbers, such as the numbers 1, 2, and 3, we use the word set. We could use set notation with braces, [ ], to list the number: [1, 2, 3]. The set of even numbers could be written as [2, 4, 6, 8, 10, …], and the set of odd numbers as [1, 3, 5, 7,…]. (The three dots indicate that the numbers continue indefinitely. In any collection of numbers ending in dots, there is no largest number.)

Here, we deal with two sets of numbers: the counting numbers [1, 2, 3, 4,…] and the whole numbers [0, 1, 2, 3,…]. The whole numbers are just the counting numbers plus zero. When we count, we start with 1. When we answer the question “How many?” we need zero as a possible answer.

Symbols are necessary to make mathematical statements complete. For example, we use symbols for addition (+) and multiplication (X).

= as in 8 + 3 = 11

8 plus 3 equals 11

< as in 3 < 8

3 is less than 8

> as in 8 > 3

8 is greater than 3

Notice that the symbols for less than and greater than are always open toward the larger number. When statements are not true, we put a slash through the symbol:

6 + 3 =/ 11

6 + 3 does not equal 11

5 >/ 7

5 is not greater than 7

9 </ 6

9 is not less than 6

Numerals are symbols for numbers, which are abstract ideas. For example, a fisherman 8000 years ago might record that he caught ||| fish. We could write 3 for the amount ||| and 3 are the symbols for the same numbers. Our number symbols are called arabic numerals.

Digits are the number symbols (numerals) 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 in our number system. Numbers are written as combinations of any of these ten digits.

A whole number is written as a string of digits, 7 is a one-digit number; 32 is a two-digit number with 3 as the first digit and 2 as the second digit; 487 is a three-digit number with 4 as the first digit, 8 as the second digit, and 7 as the third digit.

Depending on its place in the number, a digit will take on different values. For example, we read the number 67 as sixty-seven, that is, 60 plus 7, 6 is in the tens place and is worth 6 tens, or 60, while 7 is in the ones or units place and is therefore worth 7.

We read 128 as one hundred twenty-eight: 100 + 20 + 8. 1 is in the hundreds place, 2 is in the tens place, and 8 is in the units place.

2346 is read as two thousand three hundred forty six: 2000 + 300 + 40 + 6. 2 is in the thousands place, 3 is in the hundreds place, 4 is in the tens place, and 6 is in the units place.

Exercise

- Use set notation to indicate:

the counting numbers from two through seven.

the counting numbers larger than five.

the first three whole numbers.

the first five odd whole numbers. - Find the value of the digit 5 in the following numbers:

50

125

5326

5 - Translate into mathematical symbols:

Four hundred five.

Six hundred fifty.

Three thousand fifty-six.

Six thousand four hundred. - Write the following in words:

5236

8204

7029

1002