In the number system, decimals are used to express the numbers that are a minor portion of a whole number. The decimal number system is similar to fractions in the concept, but the way of representation is different. Let us consider some real-life circumstances where without our knowledge we come across decimals.
1. When you buy two candies and the bill shows you have to pay $5.68 including taxes.
2. You saved $25.5 which was given as pocket money to you from your parents.
3. A hamburger costs $2.49.
If you observe the list of scenarios mentioned above, you will notice that there is a point in between numbers. The point is termed as decimal point.
A decimal number is defined as a number with a decimal point that separates the whole number and fractional part. Decimals help us to express the quantities or values in a more precise manner. The weight of a person (184.5 lbs, 150.5 lbs) and the length of a tree (8.9 meters, 7.5 meters) can be considered as some examples of decimals. The following image describes the parts of a decimal number.
Place Value Chart of a Decimal Number
You are familiar with the place value chart of a number. The order of the place value is always considered from right to left as ones, tens, hundreds, thousands, and so on. When we move from left to right after a decimal point we have tenths, hundredths, thousandths, and so on.
Example 1: Write the place value of each digit in the given decimal number system.
a) 432.3
b) 128.67
c) 378.984
Solution:
a) In the number 432.3, 432 is a whole number part and 3 after the decimal point is a fractional or decimal part. The number is written in a place value table to understand the positions of each digit in a better way.
We read the number as the whole number part first, followed by “point” and decimal part. Number 432.3 is read as “Four hundred and thirty-two point three”. Numbers after the decimal point are always read by its face value.
b) In the number 128.67, 128 is a whole number part and 67 after the decimal point is a fractional or decimal part. The number is written in a place value chart as shown.
We read the number as the whole number part first, followed by “point” and decimal part. Number 128.67 is read as “One hundred and twenty-eight point six seven”. Numbers after the decimal point are always read by its face value.
c) In the number 378.984, 378 is a whole number part and 984 after the decimal point is a fractional or decimal part. The number is written in a place value chart as shown.
We read the number as the whole number part first, followed by “point” and decimal part. Number 378.984 is read as “Three hundred and seventy-eight point nine eight four”. Numbers after the decimal point is always read by its face value.
There is another way to read the decimal numbers, where we read the whole number part first, add “and” followed by the number after the decimal. For example, the above numbers in example 1 can be read as shown.
Types of Decimals
Decimals can be classified into the following types based on the type of digits after the decimal point:
Decimal numbers that end at a finite decimal place and do not repeat are known as terminating decimals. For example, 57.25 has terminating decimals because the number 25 does not continue after the decimal point.
Decimal numbers with an infinite number of digits following the decimal place are known as non-terminating decimals. For example, 64.25796843687… has infinite digits after the decimal point.
Depending on the pattern the decimal numbers follow after the decimal point, non-terminating decimals are further classified into recurring and non-recurring decimals.
If the digits after the decimal point repeat after every fixed interval, such numbers are called recurring decimal numbers. For example, 225.6565656565… is a recurring decimal number. Here, digits 6 and 5 repeat alternatively.
Examples of recurring decimal numbers:
1. 12.3333333….
2. 29.111111111…..
3. 67.16666666….
The recurring decimal numbers can be written using a bar line on the repeated digits. For example, 12.3333333… can be written as 12.3. A line on the digit 3 implies that 3 is a recurring decimal. Similarly, other digits can also be expressed as follows:
225.656565…… → 225.65
67.16666….. → 67.16
If the digits after the decimal point do not repeat after a fixed interval, such numbers are called a non-recurring decimal numbers. For example, 94.32578614235… is a non-recurring decimal number as the digits do not have a repeated pattern of numbers.
Expanded Form of Decimals
As we write the expanded form of numbers using its place value, we can also write the expanded form of decimals. Consider 314.68 as an example. Let us write the given number in the place value chart to understand its place values.
Now, we write the given number in the expanded form as shown.
314.68 = 3 × 100 + 1 × 10 + 4 × 1 + 6 × 1/10 + 8 × 1 /100
Example 2: Write the numbers in expanded form.
a) 359.1
b) 120.62
c) 9861.35
Solution: If you are familiar with the place value chart, you can write the expanded for directly without entering digits in the place value chart.
a) 359.1 = 3 × 100 + 5 × 10 + 9 × 1 + 1 × 1/10
b) 120.62 = 1 × 100 + 2 × 10 + 0 × 1 + 6 × 1/10 + 2 × 1/100
c) 9861.35 = 9 × 1000 + 8 × 100 + 6 × 10 + 1 × 1 + 3 × 1/10 + 5 × 1/100
Try it yourself!
Express the following decimals in the expanded form:
1. 743.68
2. 345.93
3. 180.67
4. 276.34
Math facts about the decimal number system
1. Between any two whole numbers, there are infinite decimal numbers.
2. A decimal is used to express every fraction.
3. Every decimal cannot be expressed as a fraction.
4. The winners and runners in the Olympics are determined by a minor variance in digits after the decimal point in the time.
Simon Stevin – Mathematician who applied decimals practically
Simon Stevin was a Flemish physicist, mathematician, and military engineer. His contributions in the fields of science and engineering are remarkable in both theoretical and practical aspects.
Although great mathematicians like Al-Kashi and Al-Uqlidisi had mentioned decimal fractions in their works before Stevin, Simon Stevin was the first to make its practical applications. Around the exponents of the various powers of one-tenth, Stevin printed tiny circles. The fact that Stevin used the same sign for powers of algebraic values indicates that he meant these encircling digits to denote ordinary exponents.
Know more about other concepts of Number Systems on Fibonacci Numbers, Types of Fractions, Natural Numbers, Prime Numbers and Unit Conversions.
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