At the 1984 Summer Olympics which was held in Los Angeles, U.S., P.T. Usha missed the bronze medal at the 400-metre hurdle race by only 0.01 seconds. A huge disappointment for the nation as the only chance of winning a medal in a track event was lost.
As a young student I wondered the significance of that one dot and the following numbers… To me, it meant breaking down a certain number into its constituent parts, but that very moment I realised that decimals have a far greater impact on how we perceive not just mathematics, but also life around us. That instant I thought of imagining “half” as 0.5 is no coincidence; decimals are an integral part of any habitat. Without further ado, let’s take a deeper dive into this expansive and revolutionary territory!
If you follow the decimal, from the central decimal, you can reach infinity by increasing numbers to the left of the decimal, or you can reach infinity by reducing the numbers to the right of the decimal. For example as we add to the left of the decimal point the value of the number increases vice versa as we move to the right of the decimal point the value of the number keeps decreasing as illustrated in the figure below:-.
In a strange paradox of nature, as also explained by our Yogis, whether you keep expanding or contracting a number and keep on at it, you ultimately reach the same result, i.e. you encounter infinity.
So, if we move on from any decimal to the Zero or Shunya, although the zero is empty of inherent value, it’s the zero that defines the propensity of the number. A reminder that all value come from nothingness. Zero has no properties, yet it amplifies the properties of a value placed on either side of it. Whether this number is increasing or reducing, as in 10.01, eg 100.001; 1000.0001. The dot is also the decimal or in Vedic literature it’s called the Bindu.
The decimal or the Bindu has no value.
This is what the westerners now call the point of origin of the universe in the Big Bang.
Interesting to know that the principles of decimal enumeration were already mastered in Rigveda, the oldest layer of Vedic literature. The Sanskrit terms for the nine numerals occur several times in the Ṛigveda. Among powers of ten, the Ṛigveda frequently uses das´a (ten), s´ata (hundred) and sahasra (thousand); ayuta (ten thousand), all is mentioned in a few hymns.
A more easily understandable explanation is that a decimal is used to express any and every number that is not whole. Simple!
A decimal can be thought of as the mathematical equivalent of an atom in science. Every object can be broken down further to the atomic level, just as every numeric quantity can be broken down to its smallest state with the help of a single dot placed between two numbers.
A sneak peek into the many daily life examples of decimals
1) Ravi goes to a shop to buy one pencil, but the pencil comes in a pack of two and cost is Rs 7.00 per pack. In this scenario, Ravi would simply request the shop keeper if he could buy only one pencil in this case he would simply divide 7 by 2 which is 3.5 and leave the shop with one pencil, as was his plan.
Imagine a world where the decimal did not exist… you’d simply have to buy extra items and only in whole numbers!
2) When we drive into the gas station, we pay for a certain amount of fuel and expect to receive it in full. Another way of dealing with it is to pay a certain amount and get our vehicle fuelled for that amount. Depending on the current prices of petrol etc. which might not be represented as convenient whole numbers, the volume of fuel that we receive shall also vary. This is why, when we are purchasing fuel from the gas station, we will see that we have received 12.3 l instead of a rounded-up figure.
Maintaining accurate decimal representations allow businesses to keep track of their sales without losing out on any money.
This is because accuracy is more important as every value represents something very crucial, and therefore this is yet another daily application of decimal points.
In our Ei ASSET questions too, we come across certain questions where the decimal is not understood well by the students and the students tend to round off the number and attempt the question without actually understanding it.
Let’s look at this example where the school performance showed that 82% of students were not clear on decimals as seen in the school performance and 68% of students were not clear on decimals as seen in the National Performance.
Let’s try to understand the options opted by the students:
Students generally decide which of two decimals is larger. Of course, some of these students are true experts, with a good understanding of decimal notation. This question tests the ability of students to compare decimals.
i) Students often compare decimals as whole numbers.
ii) It is important that students understand the basic concept of decimals.
iii) The numbers closest to the decimals have more weightage than those away from the decimal.
iv) Eg- 0.2 is greater than 0.1984 because 0.2 can also be written as 0.2000 which is greater than 0.1984 because the place value of 2 out weights the place value of 1 in the given example.
v) Teaching aids like Linear Arithmetic Blocks may give students a better understanding of place value and reduce errors in comparison.
vi) Help them understand that 95.18 lies in between 95.1 and 95.2.
vii) Sufficient practice in marking off decimals on a number line may be of help.
So let’s see how do we understand decimals on a Number Line:
To find out which of two decimals is the larger, append zeros to the shorter until they have the same lengths and then compare as whole numbers. For example, to compare 0.4 and 0.457, append zeros to 0.4 to get three decimal places (0.400) and then compare 0.400 with 0.457. This strategy always works for comparing decimals.
In various interactions with students it has been observed that many well-taught children correctly follow this rule, but talking to them reveals a wide range of misconceptions. They know the rule, but do not understand it. Some will forget the rule fairly quickly if it is not taught with understanding.
In the given number line, we can see how 0.01 is less than 0.1 due to the place value of “1” in both cases. We can add extra zeros to make 0.01 to 0.010 and 0.1 to 0.100. When we see the face value of 1 in both cases, we find that 0.010 can be seen as “10” in comparison to 0.100 which can be seen as “100”.
Another strategy can be Left to Right Comparison
This correct strategy is to compare columns from left to right, until a digit in one decimal is larger than the corresponding digit in the other (and the first will then be larger than the second), OR until one decimal stops (which will then be the shorter one, except in the case of zeros).
An example: to compare 23.873 with 23.86:
Money thinkers apparently have a good understanding of the first two decimal places, but are not sure of the order of other numbers on the number line.
One tertiary student, for example, when asked to place numbers between 3.14 and 3.15 on a number line did not realise that he had omitted 3.141, 3.142, 3.143, 3.144, 3.145, 3.146, 3.147, 3.148 and 3.149 which completes the number line.
The students repeatedly omitted some numbers in several similar tasks, and admitted that they were unsure of their answers. some students think that numbers such as 4.45 and 4.4502 are really equal. These students (in fact some are adults) may believe that the extra digits on the end are ‘mis-hits’ and shouldn’t really be there.
It is often very surprising how closely students’ answers follow the predictions made above across a range of tasks.
Let me conclude with a small poem which says:
Reading Decimals is easy, you will see.
They have 2 names, Like You and Me.
First, Say the Name
Without the Dot,
Then, Say the Name
Of the last place Value spot
