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This is the 4th article in the Quant Fundamental series, which covers topics from Numbers. In the first part of this article, we covered some important properties and techniques related to Fractions. In the second part of this article, we shall discuss similar things about Decimals.
In case you missed out on reading the first part, here’s the link for you:
Fractions to Decimals & back
Let’s get started then, with our discussion on the different aspects of Decimals, that can be tested on the Quant section of the GMAT.
Quick Overview of the Basics
Unlike fractions which are a category of Numbers, decimals are considered to be representations of some of these numbers. In other words, decimals are not the Number themselves, but just another way of writing the number.
As such, it’s important to understand the basic nomenclature of Decimal notation. The decimal is represented by a ‘DOT’ placed between the whole number part and the fractional part of the number. This DOT is known as the decimal point.
Most of us tend to remember the names of the digits to the left of the decimal point i.e., the whole number part – the digits are called Units, Tens, Hundreds and so on depending on the Place Value. However, many of us forget the names of the digits to the right of the decimal point i.e. the fractional part.
The first digit to the right of the decimal point is known as the Tenths digit; this is because it represents the number of 1/10ths that are there in the number.
The second digit to the right of the decimal point is known as the hundredths digit; this is because it represents the number of 1/100ths that are there in the number.
The nomenclature continues like this into thousandths, ten-thousandths and so on.
Types of Decimals
Decimals can be classified based on the number of digits after the decimal point.
If the digits stop after a certain number of places, the decimal is called a terminating decimal.
On the other hand, if the digits keep on showing up till infinity, the decimal is called a non-terminating decimal and is indicated by writing a sequence of dots after a certain number of digits.
A common mistake that many test takers make is to assume that non-terminating decimals mean recurring decimals. This is not so.
Non terminating decimals can be classified as recurring and non-recurring decimals.
Recurring decimals are so called because the digits of the decimal repeat in a pattern. The number of digits after which the pattern repeats, is known as the period of the decimal.
On the other hand, non-recurring decimals have digits coming up in random order and going on till infinity.
For example, 0.818181...... is a recurring decimal because the digits 8 and 1 repeat in the same pattern throughout. Note that this is still a non-terminating decimal, but a recurring one.
On the other hand, in the example of 0.01001000100001..... no pattern can be observed. Therefore, the decimal is a non-terminating, non-recurring decimal.
It is important to note that recurring decimals and terminating decimals always represent Rational numbers.
Non-terminating and non-recurring decimals represent Irrational numbers.
Recurring decimals could be of two types, Pure & Mixed.
Pure recurring decimals are those recurring decimals in which all the digits recur.
For example, 0.45454545…. is a pure recurring decimal since the two digits 4 and 5 both recur.
Note: 0.454545….. can be represented as 5/11
Mixed recurring decimals are those recurring decimals in which some decimals don’t recur and some digits do.
For example, 0.1666666… is a mixed recurring decimal in which the digit 1 does not recur whereas the digit 6 recurs.
So, that was about the classification of decimals. In the third and final part of this article, we will demonstrate techniques to convert terminating and recurring decimals to fractions. For now, let us understand an important technique involving fractions, which can get confusing for a lot of students.
Converting a decimal to Scientific notation
Scientific notation is the representation of a number, written using powers of 10. Therefore, when numbers which are not multiples of 10 are required to be converted to scientific notation, decimals are encountered.
For example, a number like 25430 can be written as 2.5430 * \(10^4\) or 25.430 * \(10^3\) and so on.
However, it’s not in such examples that students face difficulty. It’s mostly in questions involving smaller decimals like 0.0023 or 0.00017 and so on.
Let’s take two examples:
0.0023 = 23 * \(10^{-4}\) and 0.00017 = 17 * \(10^{-5}\).
If you did not quite understand how this happened, do not worry. Here’s how it is done.
Observe the significant digits of the number.
In 0.0023, the significant digits are 23 and in 0.00017, the significant ones are 17.
Next, observe how many digits are there after the decimal point
In the first case, there were 4 and in the second, there were 5; this will tell you the value of the exponent.
Lastly, because these decimals are less than 1, the power of 10 will be less than 0 i.e. negative since \(10^0\) = 1. So, a negative sign is affixed to the number obtained in the previous step.
Try to convert the following decimals to scientific notation:
We hope that you are now clearer about how to convert decimals to scientific notation. Go ahead, take a few decimals of your own and practice till you gain more confidence.
Rounding Decimals
Rounding decimals is not any different from rounding whole numbers. Depending on the digit to be rounded off, the digit to the right is to be observed.
For example, if a question asks you to round a decimal to its hundredth digit, this can be done on the basis of the value of the thousandth digit; similarly, if we have to round a number to its tenth digit, we look at the hundredth digit.
If the digit to the right is any one of 5,6,7,8 or 9 then the digit to be rounded is rounded up i.e. it’s value is increased by 1.
If the digit to the right is any one of 0,1,2,3 or 4 then the digit to be rounded is rounded down i.e. it’s value remains the same.
Let’s take an example to understand this better:
Say, we have to round off 0.45638 to the thousandth place.
The thousandth digit is the third digit to the right of the decimal point, therefore 6 in this case. To round off till this place, we need to look at the digit to the right of 6; in this case, it is 3. Therefore, 0.45638 will be rounded down to 0.456
Note that the thousandth digit remained 6 since the digit to the right was less than 5.
On the other hand, if we had to round off the above decimal to the tenth place, we observe that the digit to the right of 4 is 5; therefore, the decimal will be rounded up and will yield 0.5
We hope that the rounding technique is clearer now with these examples.
This brings us to the end of today’s article on Decimals. We hope that this has been useful for you to not only review your basics but also to pick up some useful techniques to solve questions on decimals.
In the third and last part of the article tomorrow, we shall discuss techniques to convert between fractions and decimals before signing off with a quick discussion on estimation and approximation in fractions and decimals.
In case you missed out on reading the first part, here’s the link for you:
Fractions to Decimals & back
Let’s get started then, with our discussion on the different aspects of Decimals, that can be tested on the Quant section of the GMAT.
Quick Overview of the Basics
Unlike fractions which are a category of Numbers, decimals are considered to be representations of some of these numbers. In other words, decimals are not the Number themselves, but just another way of writing the number.
As such, it’s important to understand the basic nomenclature of Decimal notation. The decimal is represented by a ‘DOT’ placed between the whole number part and the fractional part of the number. This DOT is known as the decimal point.
Most of us tend to remember the names of the digits to the left of the decimal point i.e., the whole number part – the digits are called Units, Tens, Hundreds and so on depending on the Place Value. However, many of us forget the names of the digits to the right of the decimal point i.e. the fractional part.
The first digit to the right of the decimal point is known as the Tenths digit; this is because it represents the number of 1/10ths that are there in the number.
The second digit to the right of the decimal point is known as the hundredths digit; this is because it represents the number of 1/100ths that are there in the number.
The nomenclature continues like this into thousandths, ten-thousandths and so on.
Types of Decimals
Decimals can be classified based on the number of digits after the decimal point.
If the digits stop after a certain number of places, the decimal is called a terminating decimal.
On the other hand, if the digits keep on showing up till infinity, the decimal is called a non-terminating decimal and is indicated by writing a sequence of dots after a certain number of digits.
A common mistake that many test takers make is to assume that non-terminating decimals mean recurring decimals. This is not so.
Non terminating decimals can be classified as recurring and non-recurring decimals.
Recurring decimals are so called because the digits of the decimal repeat in a pattern. The number of digits after which the pattern repeats, is known as the period of the decimal.
On the other hand, non-recurring decimals have digits coming up in random order and going on till infinity.
For example, 0.818181...... is a recurring decimal because the digits 8 and 1 repeat in the same pattern throughout. Note that this is still a non-terminating decimal, but a recurring one.
On the other hand, in the example of 0.01001000100001..... no pattern can be observed. Therefore, the decimal is a non-terminating, non-recurring decimal.
It is important to note that recurring decimals and terminating decimals always represent Rational numbers.
Non-terminating and non-recurring decimals represent Irrational numbers.
Recurring decimals could be of two types, Pure & Mixed.
Pure recurring decimals are those recurring decimals in which all the digits recur.
For example, 0.45454545…. is a pure recurring decimal since the two digits 4 and 5 both recur.
Note: 0.454545….. can be represented as 5/11
Mixed recurring decimals are those recurring decimals in which some decimals don’t recur and some digits do.
For example, 0.1666666… is a mixed recurring decimal in which the digit 1 does not recur whereas the digit 6 recurs.
So, that was about the classification of decimals. In the third and final part of this article, we will demonstrate techniques to convert terminating and recurring decimals to fractions. For now, let us understand an important technique involving fractions, which can get confusing for a lot of students.
Converting a decimal to Scientific notation
Scientific notation is the representation of a number, written using powers of 10. Therefore, when numbers which are not multiples of 10 are required to be converted to scientific notation, decimals are encountered.
For example, a number like 25430 can be written as 2.5430 * \(10^4\) or 25.430 * \(10^3\) and so on.
However, it’s not in such examples that students face difficulty. It’s mostly in questions involving smaller decimals like 0.0023 or 0.00017 and so on.
Let’s take two examples:
0.0023 = 23 * \(10^{-4}\) and 0.00017 = 17 * \(10^{-5}\).
If you did not quite understand how this happened, do not worry. Here’s how it is done.
Observe the significant digits of the number.
In 0.0023, the significant digits are 23 and in 0.00017, the significant ones are 17.
Next, observe how many digits are there after the decimal point
In the first case, there were 4 and in the second, there were 5; this will tell you the value of the exponent.
Lastly, because these decimals are less than 1, the power of 10 will be less than 0 i.e. negative since \(10^0\) = 1. So, a negative sign is affixed to the number obtained in the previous step.
Try to convert the following decimals to scientific notation:
- 0.002357
0.00000194
0.015789
We hope that you are now clearer about how to convert decimals to scientific notation. Go ahead, take a few decimals of your own and practice till you gain more confidence.
Rounding Decimals
Rounding decimals is not any different from rounding whole numbers. Depending on the digit to be rounded off, the digit to the right is to be observed.
For example, if a question asks you to round a decimal to its hundredth digit, this can be done on the basis of the value of the thousandth digit; similarly, if we have to round a number to its tenth digit, we look at the hundredth digit.
If the digit to the right is any one of 5,6,7,8 or 9 then the digit to be rounded is rounded up i.e. it’s value is increased by 1.
If the digit to the right is any one of 0,1,2,3 or 4 then the digit to be rounded is rounded down i.e. it’s value remains the same.
Let’s take an example to understand this better:
Say, we have to round off 0.45638 to the thousandth place.
The thousandth digit is the third digit to the right of the decimal point, therefore 6 in this case. To round off till this place, we need to look at the digit to the right of 6; in this case, it is 3. Therefore, 0.45638 will be rounded down to 0.456
Note that the thousandth digit remained 6 since the digit to the right was less than 5.
On the other hand, if we had to round off the above decimal to the tenth place, we observe that the digit to the right of 4 is 5; therefore, the decimal will be rounded up and will yield 0.5
We hope that the rounding technique is clearer now with these examples.
This brings us to the end of today’s article on Decimals. We hope that this has been useful for you to not only review your basics but also to pick up some useful techniques to solve questions on decimals.
In the third and last part of the article tomorrow, we shall discuss techniques to convert between fractions and decimals before signing off with a quick discussion on estimation and approximation in fractions and decimals.
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