Abstract:- We come across many big calculations which we want to check. Though the idea of digital roots can be used, but it is limited to integers. This paper introduces a new idea of assigning every number a characteristic value
called “Digital Values”. Every number, real or imaginary is assigned a digital value. The digital values are mostly 1, 2,3,4,5,6,7,8 or 9. These values have many interesting properties. Although in some cases we assign some other values for our convenience. The digital values can be applied to calculations to check them. They also have interesting properties in an equation (expressions involving unknown quantities) and system of equations.
Keywords:- digital values, digital roots, digital sum, digitally irrational numbers, equi-digital functions.
1 Introduction
Sometimes it is very difficult to go back and check the whole process. It happens in many calculations, while solving equations etc. The idea of digital roots may help us in some calculations. A formula for finding the digital root of an integer is given by[1] : Digitalroot[x] = 1+Mod[(x-1),9]. The digital root of addition, subtraction, multiplication and division of integers show interesting properties. But the idea is limited to integers. This paper introduces a new concept of “digital values” to overcome this difficulty. Just like in digital roots, we assign particular values for different numbers but this can be implemented for any number (real, imaginary or complex). It follows all the properties of digital roots. The paper also introduces how these digital values can help us in verifying calculations and the application of digital values in functions and equations.
2. What is digital value?
Digital value is a characteristic value assigned to a number. We will denote digital value of a number x by //x// or by dval(x).
For a natural number the digital value is same as its digital root[1]. As in digital roots, we add the different digits and repeat the process till a single digit is reached. For 1456914 the digital value will be: //1+4+5+6+9+1+4//=//30//=3.
Similarly for 563, digital value =//563//=//5+6+3//=//14//=5
2.1 Digital value of an integer
Consider the following table:
Table 1
Number Digital Value
267 6
266 5
265 4
264 3
263 2
262 1
261 9
260 8
259 7
258 6
257 5
256 4
255 3
254 2
253 1
We observe that the digital value of the natural numbers in decreasing order repeat the pattern : “9,8,7,6,5,4,3,2,1”
For 0 and negative integers also we will follow the same pattern to get the digital value i.e. digital value of 0 is 9,-1 is 8,-2 is 7,-3 is 6 and so on. A simple way to find out the digital value of a negative integer is to subtract the absolute value of the integer from 9.For e.g.
//-8// = 9 – //8// = 9 – 8 = 1
//-5647// = 9 – //5647// = 9 – 4 =5
The above results can be obtained by the general formula [1]
Digitalroot[x] = 1+Mod[(x-1),9]
Some properties of digital values:
For two integers a and b,
(1) // a + b // = // //a// + //b// //
(2) // a – b // = // //a// – //b// //
(3) // a × b // = // //a// × //b// //
(4) // // a + b // + c // = // a + // b + c // //
(5) // // a × b // × c // = // a ×//b × c // //
(6) // 9a// = 9
(7) // 8 × a // = //-a//
(8) // 9a + b // = //b//
(9) // a! // = 9, where a ? 6
(10) // a^b // = // dval(a)^b //
All the above identities can be easily proved using congruence.
2.2 Division of integers (digital values of rational numbers)
For division consider the following expression:
(11) // a/b // = // (dval(a))/(dval(b)) //
So, now, digital value for any decimal number which is terminating can be found out.
For e.g. //12.321// =// 12321/1000 // = // (dval(12321))/(dval(1000)) // = // 9/1 // = 9
For 1/11
// 1/11 // = // (dval(1))/(dval(11)) // = // 1/2 //=//0.5//=5
According to the above identity // 1/7 // and // 1/16 // should have same digital value.
So, = // 1/7 // = // 1/16 // = //0.0625// = 4
Now, for any division
// x/y // = // //x// × // 1/y // //
Division by 3,6 and 9 cannot be determined. It is either undefined or has multiple digital values.
If //a//=3, // a/3// = 1, 4, 7
If //a//=6, // a/3// = 2, 5, 8
If //a//=9, // a/3// = 3, 6, 9
If //a//=3, // a/6// = 2, 5, 8
If //a//=6, // a/6// = 1,4,7
If //a//=9, // a/6// = 3, 6, 9
If //a//=9, // a/9// = 1, 2,3,4,5, 6, 7, 8, 9
In all other cases the digital value is digitally imaginary (see next section).
2.3 Digital values of irrational numbers
For an irrational number, we will use
(12) // a^b // = // dval(a)^b //, where a, b are real numbers
So
//square root of 13// = // square root of //4// //
= //2// or //-2//
= 2 or 7
//?4 // = //2// = 2 (one root is taken only if the given value is rational)
//?13// will have 2 values : 2 and 7
Let A be another number such that //a//= //A//
// a^b // = // dval(a)^b //
and // A^b // = // dval(A)^b //=// dval(a)^b //
therefore, // a^b // =// A^b //
Using this method:
// square root of 7//= //square root of 16//= //4// or //-4//
= 4 or 5
Following is the table for digital values of some powers:
Table 2
// x^1 // 1 2 3 4 5 6 7 8 9
// x^2// 1 4 9 7 7 9 4 1 9
// x^3// 1 8 9 1 8 9 1 8 9
// x^4// 1 7 9 4 4 9 7 1 9
// x^5// 1 5 9 7 2 9 4 8 9
// x^6// 1 1 9 1 1 9 1 1 9
// x^7// 1 2 9 4 5 9 7 8 9
// x^8// 1 4 9 7 7 9 4 1 9
// x^9// 1 8 9 1 8 9 1 8 9
There is repetition in the digital values of the numbers raised to increasing powers.
For 1 : 1
For 2 : 4,8,7,5,1,2
For 3 : 9
For 4 : 7,1,4
For 5 : 7,8,4,2,1,5
For 6 : 9
For 7 : 4,1,7
For 8 : 1,8
For 9 : 9
Following this repetition digital value of any number raised to any natural power can be determined.
For e.g.
//14^11// = //5^11//=//5^5// [following the repetition]
= 2
For //x^(1/b)// , x belongs to R, b belongs to Z , a digital root between 1 to 9 exists only if it is present in the Table 2 in the row of bth power of x.
Otherwise the digital root is represented by //x^(1/b)// only. For e.g.
?3,?2
These values are called digitally imaginary numbers (DI).
2.4 Digital values of imaginary numbers
We know that
// a^b // =// A^b // when //a//= //A//
Using the above relation, when b= (1/2), a= -1, A= 8;
// i // =// ?(-1)//=//?8//
//?(-5) //=//?4//= 2 or 7 [two values because we cannot have a rational value of ?(-5) ]
OR
//?(-5) =//?5 i//=//?5.?8 //=//?4//= 2 or 7
In this way we can find the digital value of a complex number.
As in case of digital roots[2] the digital values also show the repetition in addition (Table 3), subtraction (Table 4), multiplication (Table 5)and division.
Table 3: Addition Table
+ 1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9 1
2 3 4 5 6 7 8 9 1 2
3 4 5 6 7 8 9 1 2 3
4 5 6 7 8 9 1 2 3 4
5 6 7 8 9 1 2 3 4 5
6 7 8 9 1 2 3 4 5 6
7 8 9 1 2 3 4 5 6 7
8 9 1 2 3 4 5 6 7 8
9 1 2 3 4 5 6 7 8 9
Table 4: Subtraction table
– 1 2 3 4 5 6 7 8 9
1 9 8 7 6 5 4 3 2 1
2 1 9 8 7 6 5 4 3 2
3 2 1 9 8 7 6 5 4 3
4 3 2 1 9 8 7 6 5 4
5 4 3 2 1 9 8 7 6 5
6 5 4 3 2 1 9 8 7 6
7 6 5 4 3 2 1 9 8 7
6 7 6 5 4 3 2 1 9 8
9 8 7 6 5 4 3 2 1 9
Table 5: Multiplication Table
X 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 1 3 5 7 9
3 3 6 9 3 6 9 3 6 9
4 4 8 3 7 2 6 1 5 9
5 5 1 6 2 7 3 8 4 9
6 6 3 9 6 3 9 6 3 9
7 7 5 3 1 8 6 4 2 9
8 8 7 6 5 4 3 2 1 9
9 9 9 9 9 9 9 9 9 9
3 Equality of digital values
For two equal quantities are equal the following properties of digital roots are important:
?If two quantities are equal there digital values must be equal.
This property may be used to
Check calculations:
See if digital values of both sides are equal or not. If they are not equal then the calculation is incorrect.
To find a missing digit:
Find the digital value of the known side. Then apply trial and error to put the unknown digit so that the digital values of both sides are equal.
? If a DI occurs in digital value of LHS of any equation it must occur in that of RHS too.
4. Digital value in functions and equations
In functions and equations digital values have following properties:
?For any function
(13) //f(x)// = // f (//x//) //
? In a system of equations with unique solution, the solution can be represented by an expression containing coefficients. So, if two systems of equations have equal digital values of corresponding coefficients of corresponding equations, then the corresponding roots have equal digital values.
i.e. a_11 x+ b_11 y+ c_11=0
a_12 x+ b_12 y+ c_12=0
AND
a_21 x+ b_21 y+ c_21=0
a_22 x+ b_22 y+ c_22=0
Will have same digital values of x as well as y if
//a_11//=// a_21//
//b_11//=// b_21//
//c_11//=// c_21//
?If
//a_1//=//b_1//
//a_2//=//b_2//
//a_3//=//b_3//
……………..
//a_n//=//b_n// (14)
Then
(x-a_1 )(x-a_2 )(x-a_3 )……..(x-a_n) and (x-b_1 )(x-b_2 )(x-b_3 )……..(x-b_n) are equi-digital.
The converse is not always true.
?In case of quadratic equation the converse is true when the roots are distinct.
4. Conclusion
The paper has introduced a concept of digital values which provides a way not for verifying calculations involving not only integers but any complex number. Now any complex calculation can be checked but one should be careful that if digital values of LHS and RHS are equal it does not necessarily mean that LHS = RHS. But if they are not equal then LHS cannot be equal to RHS. We have also studied the properties of digital values in functions and equations. We have also learnt how to use the property of digital value to find a missing digit in calculations.
It may seem strange to learn a way of checking a calculation when so many accurate computers are available but we must have the knowledge of the interesting properties of numbers.
References:
[1] Weisstein, Eric W. “Digital Root.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/DigitalRoot.html
[2] Teknomo,K.,”Digital Root” https://people.revoledu.com/kardi/ ,Page2
[3] Teknomo,K.,”Digital Root” https://people.revoledu.com/kardi/ ,Page7-8