Decimals make fraction math easier. They let us use the make same tools we use for whole numbers to compare and evaluate problems involving factional values and elements. They also make it easier for us to count things like money and distances. Without these numbers, we would not have the civilization we have to day. However, to properly use decimals, you have to understand where they lie on the number line. You have to know which decimal numbers are bigger than the rest so that your answers come out right.
What Are Decimals
Decimals are numbers that contain a decimal point. They can either be whole numbers or fractions, but they always have a decimal point. Even when we write whole numbers without a decimal point, the point is implied by the number system we use. Decimals are also fractions where the denominator is a power of ten. You need to know how this works to understand how decimals fit on a number line.
Fractions are ratios between two whole numbers separated by a line. The numerator is the first number and it represents the number of stuff you have while the second number, the denominator, tells us how much make the whole. They help us express values we cannot represent as whole numbers such as half (1/2) and quarter (1/4). We can even combine them with whole numbers to create mixed numbers such as 346¾, which we read as three hundred forty-five and three fourths. Decimals are just factions where the denominator is a power of ten.
Decimal numbers let us save space when writing fractions streamlining a lot of the arithmetic and math with that of whole numbers. We write these numbers as single numbers with a period. This period, called the decimal point, separates the whole number from the fractional part of the number. If we don’t have a whole number, we usually write a 0 in front of the decimal point.
The Decimal Number System
Technically, we don’t have mixed numbers when we deal with decimals. We have rational and irrational numbers instead. Rational numbers are numbers we can write as fractions. We write 346¾ as a rational number as 346.75. Irrational numbers are numbers we cannot express as fractions. Both types of numbers make up the Decimal Number System.
The decimal number system expresses numerical values as a sequence of ten symbols called digits. These digits are the familiar 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For example, we can combine the digits 5 and 7 to create the number 57. Each digit in a number represents the power of 10 we need to multiply that digit to put it in that place. The exponent of this power of ten represents the number of places between our digit and the decimal point. Whole numbers use positive exponents while the fractional part uses negative exponents.
For example, in the number 3598.726
3 is in the +3 or thousands position.
5 is +2 or hundreds position.
9 is +1 or tens position.
8 is in the 0 or unit position.
7 is in the -1 or tenths position.
2 is in the -2 or hundredths position.
6 is in the -3 or thousandths position.
These positions are the heart of the decimal number system. They show us how to read numbers and offer to compare them. 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 as we read. For example, we read the number 3598.726 as three thousand five hundred ninety-eight and seven hundred twenty-six thousandths.
Decimals on the Number Line
The decimal number system also provides for negative numbers. Negative numbers are values that are less than zero. They are the reflection of normal numbers across zero, and serve to add us in subtraction. We specify negative numbers by placing a minus 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.
With the existence of negative numbers, we refer to regular numbers as positive numbers. A positive number is any number that is neither negative nor zero. You can use a plus sign to indicate positive numbers, but we rarely do this unless we have to make the distinction Zero itself is neither positive nor negative. The decimal number system number line is a way to visualize the ranking of these numbers.
The number line is just a sequence of numbers ordered from least to greatest values. Technically, the number line is just the sequence of numbers, though we often draw a line near the number as a visual aid. We orientate the line with negative numbers on the left and positive numbers on the right. We usually only list whole numbers though we can label the line with any number sequence of our choice. 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
While whole numbers are the standard, we can include fractional values on the line as well. You have to remember that there are an infinite number of decimals between each whole number on the line.
Fractional decimals fill in the gaps between the whole numbers on the number line. We add them to the line based on the whole numbers in front of their decimal points. For instance if we expand the above number line between -2 and 2, we can add as many decimal numbers we want.
-2, -1.75, -1.5, -1.25, -1, -0.75, -0.5, -0.25, 0, 0.25, 0.5, 0.75, 1, 1.25, 1.5. 1.75. 2
I only included the quarter decimals to the line, but you can chose any decimal number you want and place it in its place on the line. You can even expand the line between any two of these decimals in order to show even more decimal numbers between them. You just have to remember that negative numbers go to the left, positive to the right, and you place the decimal according to its whole number part in the sequence. You can then read the decimals on the number line to compare them.