Decimals are used to express both the whole number and the fraction. We separate the full number from the fraction by inserting a “.”, often known as a decimal point. A decimal point (sometimes known as a decimal separator) is a point or a dot that is used to separate the entire number from the fractional part of a number.
While other countries choose a comma as a decimal separator, many students are taught to use a point. The relationship between the decimal form and fractions is critical to help students grasp decimal numbers.
What are decimals?
When writing a number that is not whole, decimals are used. Decimal numbers are numbers that fall between two whole numbers. For instance, 12.5, is a decimal number between 12 and 13. It is greater than 12, but less than 13. Decimal numbers are the same as fractions but they’re expressed differently.
Continuing with the previous example, 12.5 equals the mixed number 12 1/2. This is true regardless of how intricate the decimal is. Another example is 0.75, which is the same as 3/4. If you wanted to go even farther, you might state that 0.75 is equivalent to 75%.
Decimals are a collection of integers separated by a dot known as a decimal point. When considering the decimal place, remember that all numbers to the left of the point are whole numbers, i.e., units, tens, hundreds, and thousands. All of the numbers to the right are decimals, which are tenths, hundredths, and thousandths. It’s important to understand place value when solving mathematical problems!
Types of decimal numbers
The different types of decimal numbers include:
- Terminating decimals
- Non-terminating decimals
- Recurring decimals
- Non-recurring decimals
Terminating Decimals
The number of digits directly after the decimal point in terminating decimal numbers is limited. The number of digits following the decimal point of the ending decimal numbers can be counted so it is finite.
Here are examples of terminating decimals:
- 98.678
- 34.9
- -5.8764
All of these decimal numbers are ending decimal numbers or precise decimal numbers because the number of digits following the decimal point is finite!
Non-Terminating Decimals
Non-terminating decimal numbers are those in which the digits following the decimal point of non-terminating decimals repeat indefinitely. In other words, decimal numbers can have an endless number of digits following the decimal point. Non-terminating decimals are classified as recurring and non-recurring decimal numbers.
Recurring Decimals
Recurring decimal numbers have an unlimited number of digits following the decimal point. These numerals, however, are repeated at regular intervals.
Here are examples of recurring decimals:
- 4.33333…
- 1.54545454…
These are examples of recurring decimal numbers, in which the number of digits following the decimal point is repeated at regular intervals or in a predefined order. These numbers can also be written by placing a bar sign over the number that is repeated after the decimal point. These numbers can also be represented in fractional form, making them rational numbers.
Non-Recurring Decimals
Non-recurring decimal numbers are decimals that do not terminate and do not repeat. Non-recurring decimal numbers have an infinite number of digits at their decimal places, and their digits do not follow a fixed order.
Here are examples of non-recurring decimals:
- 3.14159265359…
- 789009.97658…
- 45.7789…
Non-recurring decimal numbers cannot be represented by a bar sign because the digits after the decimal point do not repeat in a predictable order.
What are terminating and repeating decimals?
Place value in decimals
The value of each digit in a number is given by place value. Take the number 42, for example,where 4 is worth 4 tens, or 40, and 2 is worth 2 units, or 2. The same holds true for decimals. The 2 in the number 2.78 is worth two units while the 7 is worth seven tenths, and the 8 is worth eight hundredths.
Here is a number line which is a visual way of looking at place value:
Hundreds – Tens – Units – POINT – Tenths – Hundredths – Thousandths
Using this, which of the following is greater: 2.5 or 2.15?
Both numbers have a 2 in the unit column, so move on to the next digit. This is the initial decimal place digit. The first number has a 5 in the tenth column, but the second number has a 1. Because 5 is greater than 1, 2.5 is greater than 2.15.
Hundreds – Tens – Units – POINT – Tenths – Hundredths – Thousandths
2 . 5
2 . 1 5
Properties of decimals
Here are key properties of decimal numbers:
- The product remains the same when any two decimal integers are multiplied in any order.
- The product remains the same when a whole number and a decimal number are multiplied in any order.
- When you multiply a decimal fraction by one, the result is the decimal fraction itself.
- The product of a decimal fraction multiplied by 0 is zero (0).
- The quotient of a decimal number divided by one is the decimal number.
- When a decimal number is divided by another decimal number, the quotient is one.
- The quotient of 0 divided by any decimal is 0.
Reading decimal numbers
When reading decimal numbers, read the full number section normally, then use “and” to symbolize the decimal point, then finish with the last-place value. Do not use the word “and” if there are no whole numbers in front of the decimal value. Rather, you read the number normally and end with the number’s last-place value.
Decimal to fraction conversion
It is simple to convert decimals to fractions or fractions to decimal numbers. In the following paragraphs, we go over both conversion methods in detail.
Converting Decimals to Fractions
The conversion of a decimal to a fraction is fairly straightforward. We all know that the numbers following the decimal point represent tenths, hundredths, thousandths, and so on. As a result, while converting decimals to fractions, write the decimal numbers in expanded form and simplify the values.
Using the list below, we can place our units on the line to see what they represent:
Hundreds – Tens – Units – POINT – Tenths – Hundredths – Thousandths
0 . 7 5
0.75 represents 7 tenths and 5 hundredths. As such, to form a whole number, we would have to multiply the decimal by a hundred. The point of this is that whatever your highest unit is (whether it be tenths, hundredths, or thousandths) is what the denominator will represent.
The following examples should help clarify this concept:
- 0.5 is 5 tenths, meaning as a fraction, 0.5 is 5/10
- 0.12 is 1 tenth and 2 hundredths, meaning as a fraction, 0.12 is 12/100
- 0.434 is 4 tenths, 3 hundredths, and 4 thousandths, meaning as a fraction, 0.434 is 434/1000
Once converted, you can simplify the fractions. For example, 0.2 is 2/10 which can be simplified to 1/5.
Converting Fractions to Decimals
Simply divide the numerator by the denominator to convert a fraction into a decimal. 7/2, for example, is a fraction. We then take the numerator (7) and divide it by the denominator (2) which equals 3.5. It is that simple!
You can simplify the fraction to begin with to make the division easier. For example, 20/100 can be simplified to 2/10 and further simplified to 1/5. Through division, we find that 20/100 converts to 0.2.
Rounding Decimals to the Nearest Tenths
Rounding decimals is the process of reducing a decimal number to a specific number of decimal places to save time and express large numbers in more concise terms. When knowing exact values isn’t critical, we can round decimals to the nearest wholes, tenths, or hundredths.
The value of a rounded number is about the same as the original number, but it is a little less precise. Rounding decimals is actually quite straightforward. If the last digit of the decimal ends in 5 or above then you round up (3.5 rounds up to 4.0 and 2.38 rounds up to 2.40).
If the final digit is below 5 then you have to round down (2.1 rounds down to 2.0, and 5.192 rounds down to 5.190). Always only ever round the final digit. Do not, for instance, round 2.51 to 3.00, rather 2.51 rounds down to 2.50. If you’re about to round a number and it’s followed by 5, 6, 7, 8, 9, round it up. If it’s followed by 0, 1, 2, 3, 4, the number gets rounded down.
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