Investigating the difference between 8 – 1 – 1 and 8 – 2
Particularly recall the discussions in the Learning Outcomes on the multiplication expressions 3 x 4 and 3 x 2 x 2 in the ‘Foundations of Multiplication’ book, and the expression 1 + 2 + 3 + 4 + 5 in the ‘Foundations of Addition’ book. In these Learning Outcomes, one of the emphasis was that every unique mathematical expression represents a uniquely quantified reality or imagined situation. This is an interesting and foundational appreciation of mathematics as a language. Expressions 8 – 1 – 1 and 8 – 2 are no exception.
We are using 8 – 1 – 1 and 8 – 2 as examples to discuss the details of subtraction expressions because the two are the simplest ‘3 terms’ and ‘2 terms’ expressions; by the way, 5 – 1 – 1 is not any simpler than 8 – 1 – 1 as an expression.
What is 8 – 1 – 1?
8 – 1 – 1 isn’t about being 6 (whatever we call it – difference, or ‘answer’). 6 is only one of the information contained in 8 – 1 – 1. 6 is the ‘answer’ to the query ‘what is left?’ in the expression 8 – 1 – 1, it is the quantities difference.
Let’s get what all may be inferred from 8 – 1 – 1, and here it is:
1. What action has happened to be expressed as 8 – 1 – 1?
There are two distinct, identifiable usages/giveaways/loan/investment, out of the 8 units in possession.
This information is the same in 8 – 2 – 3, 2 distinct usages of the 8 units.
Henceforth we will use the term ‘transaction’ for usage/giveaway/loan/investment to simplify the written text.
2. What quantities have been involved?
The two transactions are of 1 unit each.
This information is the same in 7 – 1 – 1, 2 distinct transactions of 1 unit each.
3. The unit of the transactions
The one unit transactions could be any quantity in 8 – 1 – 1. For example, the unit of ‘8’ and ‘1’ could be 1 apple, 1 dozen eggs, 1 km, INR 1000, or 1 apartment building
4. Unit of the 3 quantities – 8, 1, and 1 – must be the same
The unit of the ‘1 unit’ and the ‘8 units’ is the same.
If the 1 unit is 1 apple, then it implies that the 8 units are 8 apples; if the 1 unit is 1 dozen eggs, then it implies that the 8 units are 8 dozen eggs; if the one unit is 1 km, then it implies that the 8 units are 8 km; if the one unit is INR 1000, then it implies that the 8 units are 8 INR 1000 (i.e., INR 8000); if the one unit is 1 apartment building, then it implies that the 8 units are 8 apartment buildings.
5. The order of the transactions is 1 unit and then 1 unit
While it is not clear in 8 – 1 – 1 example, but in the 8 – 2 – 3 example, the order of transactions is 2 units and then 3 units.
6. The nature of the 1 unit – the amount of transactions (the subtrahend)
The two transactions of 1 unit each do tell us that the nature of the 8 quantities maybe (yes, we can’t be sure, but we do know this to a level) such that 1 unit of the 8 units is good enough, or common, quantity of usage/giving away.
For example, suppose 8 units is 8 INR 10. In that case, we can appreciate that its transactions (usages) can be in terms of 1 INR 10 each; INR 10 can buy many kinds of small cookie packets, chocolates, etc.
Broadly, we mean that the transactions of a quantity of something are related to the quantity's nature and value. Moreover, the quantity of transactions does tell us something about the thing itself (at least over a larger number of transactions).
7. The nature of the 8 units – the total quantity (minuend)
Quite similarly, the quantity of the minuend does tell us a lot about the nature of the minuend. Of course, we cannot say much about the nature of the minuend with high confidence based on the knowledge of quantities used for the minuend in a few expressions. However, we can expect some definite ideas about the minuend even from one expression.
For example, 8 INR 1 – 1 INR 1 – 1 INR 1 isn’t a realistic subtraction expression, as compared to 8 USD 1 – 1 USD 1 – 1 USD 1 is more realistic because one can have a decent meal for two in USD 8, but INR 8 is no money.
Indeed, there is a lot hidden in every expression of quantity!
8. 8 – 1 – 1 as an investment statement
When subtraction expressions are used to represent investments (subtrahends) from a corpus of money (minuend), the 8 – 1 – 1 reflects a less risky investment of 2 units out of the 8 units. Making 2 different investments of 1 unit each is less risky than one investment of 2 units, for instance.
Strictly speaking, 1 unit and 1 unit does not necessarily mean 2 separate investments. However, it does indicate that even in that situation, the investment is made one after the other (indicating the second investment to be thoughtfully based on the first investment).
For instance, if one has USD 800,000 to invest, then a safer strategy is to make two investments of USD 100,000 each, rather than one investment of USD 200,000. It is common sense.
9. 8 – 1 – 1 as a consumption statement 2 units of consumption (1 + 1)! Yes, this is the addition of all the subtrahends.
Later we will discover the mathematical significance of finding the sum of the multiple subtrahends as a necessity in analysing subtraction expressions. To get the total amount of consumption of a stock of things, we must add the individual consumption levels.
10. What is left?
6! A figure we all know.
What is 8 – 2?
Let’s get what all may be inferred from 8 – 2, and here it is:
1. What action has happened to be expressed as 8 – 2?
There is just one distinct, identifiable transaction out of the 8 units in possession.
This information is the same in 8 – 4 also, 1 distinct transaction.
2. What quantities have been involved?
The one transaction is of two units.
This information is the same in 7 – 2 also, 1 distinct transaction of two units.
3. The unit of the transactions
The two unit transactions could be any quantity. For example, it could be two 1 apple, two 1 dozen eggs, two 1 km, 2 INR 1000, 2 apartment buildings
4. Unit of the 2 quantities – 8 and 1 – must be the same
The unit of the ‘two units’ and the ‘8 units’ is the same
If the one unit is 1 apple, then it implies that the 8 units are 8 apples; if the one unit is 1 dozen eggs, then it implies that the 8 units are 8 dozen eggs; if the one unit is 1 km, then it implies that the 8 units are 8 km; if the one unit is INR 1000, then it implies that the 8 units are 8 INR 1000 (i.e., INR 8000); if the one unit is 1 apartment building, then it implies that the 8 units are 8 apartment buildings.
5. The transaction is once and of 2 units
6. The nature of the 1 unit – the amount of transactions (the subtrahend)
The one transaction of 2 units does tell us that the nature of the 8 quantities may be such that 1 unit of it is not good enough, or common, quantity of transaction.
7. The nature of the 8 units – the total quantity (minuend)
Quite similarly, the quantity of the minuend does tell us a lot more about the nature of the minuend. Once again, we cannot say much about the nature of the minuend with high confidence based on the knowledge of quantities used for the minuend in a few expressions. However, we can expect some definite ideas about the minuend even from one expression.
8. 8 – 2 as an investment statement
Following the discussion on this topic for expression 8 – 1 – 1, we need only say that 8 – 2 is generally an expression of more risky investment than 8 – 1 – 1.
9. 8 – 2 as a consumption statement
2 units of consumption.
10. What’s left?
6! A figure we all know.
Hopefully, the aforementioned analysis provides a deeper understanding of subtraction expressions.
Summary
8 – 1 – 1 and 8 – 2 carry quite a few distinct information about the situations they represent.Excerpted from the book ‘Foundations of Subtraction (Mathematics as a language)’ by Sandeep Srivastava and Saloni Srivastava