## Some basic ideas of arithmetic

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

1. 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.
2. Find the value of the digit 5 in the following numbers:
50
125
5326
5
3. Translate into mathematical symbols:
Four hundred five.
Six hundred fifty.
Three thousand fifty-six.
Six thousand four hundred.
4. Write the following in words:
5236
8204
7029
1002
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