Eons ago, our ancestors the number of fingers they had and decided to form a numbering system around that number. This numbering system will express the values they need to record such as how much food they had and how much did it cost. This is how the Decimal Number System came to be. We continue to use it today because of how useful it is.
The Decimal Number System
The Decimal Number System represents numerical values are a sequence of symbols called digits. These digits are the familiar 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Because it has only ten digits, the Decimal Number System is also called the Base 10 Number System. We can express any numerical value with a number from the Decimal Number System.
We call any number from the Decimal Number System a decimal number or decimal. We also call decimals “numbers” due to how familiar we are with the System. While you may consider only numbers that have a period in them as decimals, every number in the decimal system is technically a decimal including whole numbers. The period, called the decimal point, just identifies where we switch from powers of ten to fractional powers of ten.
Each digit in a number represents the different power of 10 we need to multiply that digit in order to put it into its place in the number. To the left of the decimal point, we use positive powers to represent position. To the right, we use negative numbers.
For example, in the number 3598.726
3 is in the +3 position from the decimal point as its power of ten is 103 or 1000.
5 is +2 with 102 or 100 as its power of ten.
9 is +1 with 101 or 10 as its power of ten.
8 is in the 0 position with 100 or 1 as the power of ten.
7 is in the -1 position with the power of ten 1/10 or 10-1.
2 is in the -2 position with the power of ten 1/100 or 10-2.
6 is in the -3 position with the power of ten 1/1000 or 10-3.
These powers of ten positions are everything in the decimal number system. All arithmetic using these numbers must properly respect them in our answers.
Digit place (or position) is so important in the Decimal Number System that each position has a name. These names come from the powers of ten and serve to remind us of these powers. For instance, in 3598.726, the 9 is in the tens position while 2 is in the hundredths.
These potion names also help up know how to read numbers. We generally read the whole number part as is. We read them by saying each digit and then its position in the number. This continues with the decimal part with a few extra rules. Decimals are read like they were whole numbers with the name of the rightmost position. The word “and” denotes the location of the decimal point when we read a number. Using the Decimal naming scheme, we read the number 3598.726 as three thousand five hundred ninety-eight and seven hundred twenty-six thousandths.
Fractions, Rational, and Irrational Numbers
The Decimal Number System uses sequences of digits to represent values, but that is not the only way we can represent values in the system. The Decimal Number Systems allows for things called fractions.
Fractions are numbers comprised as two whole numbers separated by a line. The first number is called the numerator and represents the value we have. The second number, the denominator, represent how much we need to make the whole. The decimal part of a decimal number is the numerator of a fraction where the denominator is a power of ten.
While we can write any fraction as a decimal, there are some decimal numbers we cannot write as fraction. We call these numbers irrational and they include things like pi. All other numbers are called rational numbers.
Negative Numbers and the Decimal Number Line
The Decimal Number System also provides for a set of negative numbers. Negative numbers are values that are less than zero. We denote negative numbers by inserting a – sign in front of the number. For instance, -54675.435 is a negative decimal number. We read negative decimals by saying the word negative in front of it. We read -54675.435 as negative fifty-four thousand six hundred seventy-five and four hundred thirty-five thousandths.
Any decimal number that is not negative is either zero or a positive decimal number. We usually don’t add anything to indicate a positive number. However, we can use the + sign when we must make the distinction. Zero, 0, is neither positive nor negative.
The decimal number system number line is a way to visualize the ranking of decimal numbers. It lists all decimal numbers from lowest to highest either from left to right of bottom or top. Because, there are an infinite number of decimal numbers we usually just mark the whole numbers and a few key fractions (if needed) when we draw the line. For example, the following is the number line from -10 to 10.
-10, -9. -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
The Decimal Number System is the number system we use to express values in our daily lives. Also called the Base 10 Number System, it expresses every number as a power of 10. However, it is not the only number system we use. For instance, computers use the Base 2 System, or Binary Number System, which only has two digits, 0 and 1, and expresses values as powers of two. We use these alternative number systems to express values that would otherwise be too complicated to express as decimals, such as the on off states of an electric switch. Regardless of the number system we use, they all do one thing; help us express the values we see in the world.