Decimal Numbering System

The decimal numbering system uses ten symbols called digits, each digit representing a number. These symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The symbols are known as the numbers zero, one, two, three, etc. By using combinations of 10 symbols, an infinite amount of numbers can be created. For example, we can group 5 and 7 together for the number 57 or 2 and 3 together for the number 23. The place values of the digits are multiples of ten and given place titles as follows:

Numbers in the decimal system may be classified as integers or fractions. An integer is a whole number such as 1, 2, 3, . . . 10, 11, . . . A fraction is a part of a whole number, and it is expressed as a ratio of integers, such as 1/2, 1/4, or 2/3.

An even integer is an integer which can be exactly divided by 2, such as 4, 16, and 30. All other integers are called odd, such as 3, 7, and 15. A number can be determined to be either odd or even by noting the digit in the units place position. If this digit is even, then the number is even; if it is odd, then the number is odd. Numbers which end in 0, 2, 4, 6, 8 are even, and numbers ending in 1, 3, 5, 7, 9 are odd. Zero (0) is even.

Examples:

Determine whether the following numbers are odd or even: 364, 1068, and 257.

Solution:

1. 364 is even because the right-most digit, 4, is an even number.

2. 1068 is even because the right-most digit, 8, is an even number.

3. 257 is odd because the right-most digit, 7, is an odd number.

Adding Whole Numbers

When numbers are added, the result is called the sum. The numbers added are called addends. Addition is indicated by the plus sign (+). To further explain the concept of addition, we will use a number line to graphically represent the addition of two numbers.

Example: Add the whole numbers 2 and 3.

Solution: Using a line divided into equal segments we can graphically show this addition.

Starting at zero, we first move two places to the right on the number line to represent the number 2. We then move an additional 3 places to the right to represent the addition of the number 3. The result corresponds to the position 5 on the number line. Using this very basic approach we can see that 2 + 3 = 5. Two rules govern the addition of whole numbers.

The commutative law for addition states that two numbers may be added in either order and the result is the same sum. In equation form we have:

a+b=b+a (1-1)

For example, 5 + 3 = 8 OR 3 + 5 = 8. Numbers can be added in any order and achieve the same sum.

The associative law for addition states that addends may be associated or combined in any order and will result in the same sum. In equation form we have:

(a+b)+c=a+(b+c) (1-2)

For example, the numbers 3, 5, and 7 can be grouped in any order and added to achieve the same sum:

(3+5)+7=15 OR 3+(5+7)=15

The sum of both operations is 15, but it is not reached the same way. The first equation, (3 + 5) + 7 = 15, is actually done in the order (3 + 5) = 8. The 8 is replaced in the formula, which is now 8 + 7 = 15.

The second equation is done in the order (5 + 7) = 12, then 3 + 12 = 15. Addition can be done in any order, and the sum will be the same.

When several numbers are added together, it is easier to arrange the numbers in columns with the place positions lined up above each other. First, the units column is added. After the units column is added, the number of tens is carried over and added to the numbers in the tens column. Any hundreds number is then added to the hundreds column and so on.

Example:

Add 345, 25, 1458, and 6.

Solution:

When adding the units column, 5 + 5 + 8 + 6 = 24. A 4 is placed under the units column, and a 2 is added to the tens column.

Then, 2 + 4 + 2 + 5 = 13. A 3 is placed under the tens column and a 1 is carried over to the hundreds column. The hundreds column is added as follows: 1 + 3 + 4 = 8.

An 8 is placed under the hundreds column with nothing to carry over to the thousands column, so the thousands column is 1. The 1 is placed under the thousands column, and the sum is 1834. To verify the sum, the numbers should be added in reverse order. In the above example, the numbers should be added from the bottom to the top.