Introduction to Decimals
In the world of math, we are familiar with numbers like- real numbers, natural numbers, whole numbers, rational numbers, and more. But today, we will spotlight on decimals. They’re the numbers we use to deal with parts of things, not just whole numbers. In this post, we’re diving into decimals. We’ll cover what they are, and how they work, and even show you some cool examples to make things crystal clear and help you understand better.
Some More Concept
Decimals show parts of a whole number by using a dot to separate the whole from its fractions. In Math, decimals are a special kind of number. They’ve got two parts: a whole number part and a fraction part, and they’re split by a dot (.),we call the decimal point. Like, take 23.4 for instance – it’s a decimal number. Here, 23 is the whole number, and 4 is the fraction part. And that little dot(.), that’s the decimal point.
Types of Decimal Numbers
Terminating Decimal
Terminating decimal is the decimal that comes to an end, meaning it have a finite number of digits after the decimal point. In other words, it stops or terminate after a certain point. For instance:
- 750.75 – Here, the decimal stops after the 75, making it a terminating decimal.
- 253.25 – Similar to the first example, the decimal ends after 25, so it’s also terminating.
- 52.5 – Again, the decimal stops right after the 5, making it a terminating decimal.
In terminating decimals, you won’t find any repeating patterns after the decimal point; they simply stop at a certain digit.
Non-terminating recurring decimal
Non-terminating recurring decimal is decimal that continue indefinitely after the decimal point, and have a repeating pattern of digits. This pattern can be single digits or multiple digits that repeat predictably. Here are a few examples:
1. 0.333…0.333… – This is the decimal representation of the fraction 1/3. It repeats the digit 33 infinitely.
2. 0.142857142857…0.142857142857… – This represents the fraction 1/7. Notice the repeating pattern 142857142857, which goes on indefinitely.
3. 0.1666…0.1666… – This represents the fraction 1/6, and the digit 66 repeats infinitely.
In non-terminating recurring decimals, the pattern repeats without end, unlike terminating decimals which have a finite number of digits after the decimal point.
Remark: Terminating and Non-terminating Decimals are rational in nature
Non-terminating Non-recurring decimals
Non-terminating non-recurring decimals are decimals that continue indefinitely after the decimal point, but they do not have a repeating pattern of digits. Instead, the digits appear in a random sequence without repeating. These decimals are also called as irrational numbers.
Here’s an example:
0.235711131719…
The digits 235711131719 continue indefinitely without repeating any predictable pattern.
Other examples of non-terminating non-recurring decimals include the square root of non-perfect squares, such as √2 , √3 , √5, π, etc.
These decimals go on forever without repeating a pattern.
Remark: Non-terminating non-recurring decimals are irrational.
Comparison of Decimals
Comparing decimals involves determining which decimal is greater, less than, or equal to another decimal. Here’s how to compare decimals:
1. Compare Whole Numbers: Start by comparing the whole number parts of the decimals. The decimal with the larger whole number is greater.
Example: 2.7 is greater than 1.91 because 2>1.
2. If Whole Numbers Are Equal: If the whole number parts are equal, compare the first decimal place. The decimal with the larger digit in the first decimal place is greater.
Example: 3.232 is greater than 3.131 because 2>1.
3. If the First Decimal Places Are Equal: If the first decimal places are equal, compare the second decimal places, then the third, and so on until a difference is found.
Example: 3.253 is greater than 3.213 because in the second decimal place, 5>1 .
4. If One Decimal Ends: If one decimal ends before the other, but they are equal up to that point, the decimal that ends is considered smaller.
Example: 1.351 is greater than 1.35 because 1.351 goes one place further.
Remember, when comparing decimals, always start from the left and work your way to the right, comparing digits place by place until a difference is found.
Operation of Decimals
Decimal operations involve addition, subtraction, multiplication, and division with numbers represented in decimal notation. Here’s a brief overview of each operation:
Addition and Subtraction
Addition and Subtraction: When adding or subtracting decimals, align the decimal points and then add or subtract as you would with whole numbers. Make sure to carry any extra digits over if needed as indicated in the following figure.
Multiplication
Multiplying decimals is similar to multiplying whole numbers, but you have to consider the placement of the decimal point. Here’s a step-by-step guide:
Ignore the decimals initially: Treat the numbers as if they are whole numbers, and multiply them together.
Count the total number of decimal places in both numbers.
Place the decimal point in the result so that the total number of decimal places is equal to the sum of the decimal places in the original numbers.
Let’s illustrate with an example:
Say you want to multiply 3.5 by 2.7
Ignore the decimals initially: Multiply 35 by 27
35×27=945
Count the total number of decimal places: Both numbers have one such place.
Place the decimal point: Since there are two decimal places in total, the result will have two such places. So, the answer is 3.5 x 2.7 = 9.45
Division
Dividing decimals follows a similar process to multiplying, but with a couple of additional steps. Here’s how you can divide decimals:
Move the decimal points: First, adjust the decimal points in both the dividend (the number being divided) and the divisor (the number you’re dividing by) so that the divisor becomes a whole number. Move the decimal point in both numbers the same number of places to the right until the divisor becomes a whole number. Remember to perform the same operation on both the dividend and the divisor to keep the value of the expression unchanged.
Perform the division: Divide the adjusted dividend by the adjusted divisor as you would with whole numbers.
Place the decimal point: After obtaining the quotient, place the decimal point in the quotient so that it aligns with the decimal point in the original dividend.
Let’s illustrate with an example:
Suppose you want to divide 5.25 by 0.75.
Move the decimal points: Multiply both the dividend and the divisor by 100 (because there are two decimal places in 0.75) to make the divisor a whole number. So, 5.25×100=5255.25×100=525 and 0.75×100=750.75×100=75.
Perform the division: Divide 525 by 75.
525÷75=7
Place the decimal point: Since there were two decimal places in the original numbers, place the decimal point in the quotient so that it lines up with the decimal point in the original dividend.
5.25÷0.75=7
So, 5.25÷0.75=7.00
FAQ
- A decimal is a way of expressing numbers that are not whole. It includes a decimal point, which separates the whole number part from the fractional part.
- To read a decimal aloud, say the whole number part followed by “point” and then say each digit after the decimal point individually. For example, 3.25 is read as “three point two five.”
- To add or subtract decimals, align the decimal points and then perform the addition or subtraction as you would with whole numbers. Make sure to place the decimal point in the result directly below the decimal points in the numbers being added or subtracted.
To multiply or divide decimals, ignore the decimal points initially and perform the multiplication or division as if the numbers were whole numbers. Then, count the total number of digits to the right of the decimal point in the original numbers and place the decimal point in the result to match that count.
- A terminating decimal is a decimal that ends, such as 0.5 or 1.75. A repeating decimal, on the other hand, is a decimal that has a repeating pattern of digits after the decimal point, such as 0.333… (which can be written as 0.3 with a bar over the 3) or 0.121212… (which can be written as 0.12 with a bar over the 12).
- To convert a fraction to a decimal, divide the numerator (top number) by the denominator (bottom number). If the division results in a terminating decimal, you’re done. If it results in a repeating decimal, you can either write the repeating part with a bar over it or round the decimal to a certain number of decimal places.
- To compare decimals, start by comparing the digits to the left of the decimal point. If they are different, the number with the larger digit is greater. If they are the same, move to the next digit to the right until you find a point of difference.
