The Development of Arithmetic: Part 2*
By Anantanarayanan Thyagaraja
Renaissance thinkers had the idea of inventing extensions of the Natural Numbers. The answer to, “what number added to 2 gives 6?” is in: u+2=6; u=4 [u for “unknown”]. A similar equation, u+6=4, is “insoluble” if we only had 0,1,2,3,..to play with. Some ancient genius invented a “number” called “-2” and added it to 6, thus, 6+(-2)=4; thus “6 steps forward and 2 backwards is 4 forwards”! The negative numbers, -1,-2,-3,.. are mirror images of positive numbers and solve “insoluble” equations like u+10=0! Many consequences result: m+(-m)=0 and 0-(-m)=2 for any number.! Natural Numbers are thus extended: -5,-4,-3,-2,-1,0,1,2,3,4,5,.. . Obviously Accountants use “credit(+)” and “debit(-)” exactly like this.
The notion of repeated additions led Hindus to Multiplication of extended Natural Numbers, termed “integers”. It was noted that sets can be combined in a new way: suppose A,B,C all have equal cardinal numbers, say 2. Putting them together, we have a set with 2+2+2=6 members. This suggested: “three sets of two members is “pairable with” a set with 6 members”! Why not write, 3 x 2=6?” From here on we use “x” as a symbol for the act of multiplication, not as a number. This concept led to many deep rules: if m and n are any two positive numbers, m x n=n x m, and m x (n+p)=(m x n)+(m x p). In verbose form: “the result of multiplying m by the number (n+p) will be the same as multiplying m and n first and adding it to the result of multiplying m by p”. How wordy this is compared to the compact notation! In Mathematics, good notation is almost as important as the concepts, since it promotes efficient calculation. The familiar multiplication tables result using repeated additions. Many people bemoan the failure of some children to learn “times tables” by rote, quite forgetting that understanding and correct application of Arithmetic is far more important. The concept of multiplication implies a “Law of Arithmetic” which states: given any three integers m ,n, p: m x (n x p)= (m x n) x p. It might seem that these rules are trivial. After thousands of years the Hindu [and later Renaissance] mathematicians proved that these rules apply generally to all integers, positive or negative [or zero], showing that, (-m) x (n)= -(m x n) and (-m) x(-n)=m x n, where m, n are any two integers. The proof of this “Rule of signs” is left as a challenge to the interested reader!
The developers of Arithmetic could solve equations like, 2 x u=20 find that u=10. When Aditi’s descendant Patagonia asked her father Paleogoras, “Father, how can I solve, 2 x u =3!” Paleogoras committed suicide, having failed to solve it! Patagonia’s son, Bhasker observed: “We can divide A bag of 4 nuts exactly in half; 2 x u=4, and u=2. Let’s invent a new “number” which “fractures” 3 into 2 parts and call it the fraction, 3 divided by 2, 3/2 for short”. Thus, (3/2) has the simple property, 2 x (3/2)=2. The Hindus and Chaldeans understood 1/n to represent dividing a cake into n equal parts and hence n x (1/n)=1.
The equation 3 x u=6, has the solution u=2. What about 2 x u=13? There no integer which satisfies the equation which is an “impossible/insoluble equation”, if we require u to be an integer. The same applies to m x u=1, where m is any integer greater than 1. Bhasker, Hindus, and Chaldeans invented a new kind of number to solve this equation. These new numbers were called “fractions” or, more obscurely, “rational numbers”. They wrote “u=1/m” stating that m x (1/m)=1. Consistency then required that this “number” must also obey all the rules of Arithmetic. Evidently if m,n are any two positive integers, we can give a meaning to the equation, n x u=m, in terms of the fraction 1/n and conclude that u=m x (1/n) and simply write u=m/n.
There remains one truly impossible/contradictory equation: 0 x u=m, when m is not zero(why?)! There are special names for u, n, m: n is called the denominator, m the numerator and u the quotient/fraction. Every integer is also a fraction [the denominator is 1]. Division is the inverse of multiplication and fractions are defined by equations they satisfy, just like negative integers.
If u is a solution of n x u=m and p is any integer [other than zero!], (p x n) x u=(p x m); thus, u=(p x m)/(p x n). Multiplying the numerator and denominator of a fraction does not change it; apparently trivial, but crucial! Consider, n x u=m and q x v=p, where m,n,p, q are any set of integers [denominators, n, q must be non-zero]. The equation, (n x q) x w=(m x p), is solved by noting, w=u x v=(m x p)/(n x q)! This is just the rule for “multiplying” any two fractions, u=(m/n) and v=(p/q). This clearly “reduces” to the ordinary multiplication of integers [positive, negative and zero, when n=p=1, and corresponds precisely to our intuitions about dividing pizzas or cakes.
Lilavati wanted to add two fractions, 1/2 and 1/3. Her clever solution: “Obviously, 1/2= (3×1)/(3×2)=3/6 and 1/3=(2×1)/(2×3)=2/6. So, (1/2)+(1/3)=(3/6)+(2/6)=5/6!” Her key insight: n x u=m, qxv=p. Then (q x n) x u =(q x m), so (qxn)x(u+v)=(qxm)+(nxp). To add two fractions with the same denominator, simply add the respective numerators and use the “common denominator”! If the denominators are unequal in the two fractions, “equalise” by multiplying them and add the “equivalent” fractions! The rules of Arithmetic were then extended consistently [a crucial requirement of Mathematics!] to positive and negative rational numbers.
Mathematicians proceeded to “solve impossible equations” by extending “positive integers” and associated Arithmetical rules consistently. Practically all human tasks requiring computations are then reducible to Arithmetic. Yet, profound ideas are implicit in elementary Arithmetic, as will be described.
*part 1 was published in Confluence August 2016 issue
Mr. Anantanarayanan Thyagaraja is a theoretical physicist who was educated in Madras and in California. He worked for over thirty years at the Culham Science Centre and now in retirement continues to work in research in the U.K. and in India.