In the chapter, Unit, Number and Numeral, we learnt how to write counts of a thing using unit and numeral. Let’s take it to the next step, but how much a thing, in other words, is it more or less in numbers. So how do we compare numbers if it is bigger than the other. Here, comes the concept of Face value and Place value, that helps in understanding the number system in mathematics. Eventually, it clearly helps to know which is cheaper a watch with price tag of $20 or $200.
This is not the end, we do numerous mathematical operations on numbers for example addition, subtraction, multiplication and division on numbers. The process involved in handling these arithmetic operations are based on understanding of these basic concept of place value.
To write any number bigger or shorter we use digits. There are 10 digits in the number system which helps us to write any number in mathematics. When we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to represent any numbers, these ten symbols are called Digits or Figures.
In above example which is cheaper a watch with price tag of $20 or $200, to compare the two numbers 20 and 200 we should learn before all how the numbers are written using place value and face value. In next step, let’s learn face value and place value by seeing examples.
Face Value of a digit in a number is equal to the same digit itself and it is also known as true value. The face value of a digit is always equal to the digit itself.
Consider a number 34872
Face value of 3 in number 34872 is 3.
Face value of 4 in number 34872 is 4.
Face value of 8 in number 34872 is 8.
Face value of 7 in number 34872 is 7.
Face value of 2 in number 34872 is 2.
Place Value is a value of digit in a number and it is also known as local value. For example, the place value of digit 2 in number 34872 is 2, the place value of 7 is 70 and place value of 8 is 800. You can notice as we move from left to right in 34872 we add that much number of zeros behind the digits.
To find place value of a digit in a number, we always move from right to left. The first position on right is called 1s, and read as ones. The second position is 10s and read as tens. The next positions go as 100s then 1000s and so on.
Let’s start moving from left to right in number 34872. As we move, we keep on increasing number of zeros by one behind the digits.
So, no zero behind 2. One zero behind 7, Two zeros behind 8. Three zeros behind 4. Four zeros behind 3.
Or we can write it as multiply the digits with 1s, 10s, 100s, 1000s and 10000s depending upon their position:
2 is at 1s place, so 2 x 1 = 2
7 is at 10s place, so 7 x 10 = 70
8 is at 100s place, so 8 x 100 = 800
4 is at 1000s place, so 4 x 1000 = 4000
3 is at 10000s place, so 3 x 10000 = 30000
So, we get the place value of each digit in 34872 as 2 of 2 , 70 of 7, 800 of 8, 4000 of 4 and 30000 of 3.
In general we can find the place value of digit at a position by multiplying the face value by the value of place.
Place Value = Face Value X Value of Place
Place Value of \(4\) in number \(34872\) is calculated as:
\(4 \times 1000 = 4000\), where face value = 4 and value of place = 1000
If a digit appears more than one time in a number, the place value of that number will be different and depends upon its position in the number. Let’s understand it with an example.
Consider a number 348742, 4 repeats two times at 10s and 10000s place.
Place value of 4 at 10s place = 4 x 10s = 40.
Place value of 4 at 10000s place = 4 x 10000s = 40000.
Therefore, we can see the repeating digit 4 has two different place values 40 and 40000 in the same number 348742.
Place value helps us in comparing two or more numbers to check which is the greatest number or which is the smallest number.
In general, we are aware that the number which have more number of digits than the other number is greater than the other number. This is a good thumb point of rule to compare two numbers which have different number of digits.
Two watches with price tag of $20 or $200 has different number of digits.
20 has two digits 2 and 0. 200 has three digits 2, 0 and 0. So, 200 is greater than 20 because 200 has more number of digits than 20 and that makes $200 watch costlier than $20 watch.
Above we learnt, how to compare two numbers with unequal number of digits. The same is not the case when the number of digits in two numbers are equal. We check such numbers using place value of the left most number. If the left most number’s place value is greater than the other then the number is greater also than the other number.
Consider two numbers with equal number of digits, i.e. 3, as 246 and 514.
Step1: Take the left most digit of 246 which is 2.
Step2: Find the place value of 2, which is 2 x 100s = 200
Step2: Take the left most digit of 514 which is 5.
Step3: Find the place value of 5, which is 5 x 100s = 500
Step4: Compare 200 and 500, which is greater? It is 500.
Step5: That means 514 is greater than 246.
In above example of two numbers 246 and 514, the left most digits are different i.e. 2 and 5. There can be a task to compare two numbers with equal number of digits and where the left most digits are same.
Consider two numbers 246 and 214 which have equal number of digits i.e. 3 and have same the left most digit i.e. 2 in both numbers.
If we follow the above steps to compare them then both have 200 place value for the left most digit 2, because 2 is at 100s place and 2 x 100 = 200.
In such kind of numbers where left most digit is same, we move to next right number and see which is greater using the above method.
Here, 4 is the next right number in 246 and 1 is the next right number in 214. Both 4 and 1 are at 10s place.
So, place value of 4 in 246 is 4 x 1 = 40 and place value of 1 in 514 is 1 x 10 = 10.
So, 40 is greater than 10. That means 246 number is greater than 214.
8 occurs at ten thousand's (10000) place.
Therefore, its place value = 8 x 10000 = 80000
Face value of 5 in 532798 is 5.
The place value of 8 = 8 x 1000 = 8000
Face value of 8 = 8.
Difference between place value and face value = 8000 - 8 = 7992
Face value of 4 = 4 and place value of 4 = 4 x 1 = 4.
Face value of 6 = 6 and place value of 6 = 6 x 10 = 60.
Face value of 0 = 0 and place value of 0 = 0 x 100 = 0.
Face value of 3 = 3 and place value of 3 = 3 x 1000 = 3000.
Face value of 8 = 8 and place value of 8 = 8 x 10000 = 80000.
Face value of 5 = 5 and place value of 5 = 5 x 100000 = 400000.
So, digits 4 and 0 have their same place value and face value.
First 9 occurs at ten's place. So, place value of first 9 is 9 x 10 = 90.
Second 9 occurs at thousand's place. So, place value of second 9 is 9 x 1000 = 9000.
Face value of 9 = 9.
Place value of 5 = 5 x 1000 = 5000.
Sum of place value of 9 and 5000 = 9 + 5000 = 5009.