Digital roots, and what survives repeated digit-summing

What to assume they know

• The digit-sum test for nine, and the run-of-nines split of place values (The digit-sum test for 9, and the algebra underneath it)
• The digit test for three (Why the same digit sum also settles divisibility by 3)
• Remainders, and the fact that a remainder is smaller than the divisor
• Multiples and factors
• Letter-numbers, and recognising when a term of an expression is a multiple of a given number

What they should be able to do

• Compute the digital root of a number by summing digits repeatedly, and say when to stop
• Explain why the process always terminates
• Explain why summing digits leaves unchanged how far a number sits past a multiple of nine
• State what the digital root reports, and account for the value taken by multiples of nine
• Predict the digital roots of a run of consecutive numbers, and describe the cycle
• Predict the digital roots of consecutive multiples of a given number, and of numbers a fixed amount past a multiple
• Read the digital root of an expression off its parts, when some parts are multiples of nine
• Use digital roots to check an arithmetic calculation, and say what such a check can and cannot establish

Where it usually goes wrong

• "The digital root of a multiple of nine is zero." It is nine. The process can only stop on a digit it can actually reach by adding, and it never produces zero from a positive number. Nine and zero are the same position past a multiple of nine; nine is the representative this procedure hands you.
• "The digital root is the remainder." It reports the remainder for eight of the nine possible positions and reports nine where the remainder is zero. The chapter's two printed sentences, on Part I p.125 and p.130, say each half; a student who reads only one will be wrong a ninth of the time.
• "You can stop summing whenever the number looks small." You stop when one digit is left and not before. A student who stops at 11 or 29 has an intermediate value, not a root.
• "Bigger numbers have bigger digital roots." The root is one of nine values whatever the number's size. The eight-digit number in exercise 1 is stated to be eight digits precisely so the student can discover the digit count does not matter.
• "Rearranging a number's digits changes its digital root." It cannot — the digit total is unchanged. This links back to the reversal conjecture on Part I p.132.
• "Adding ten adds one to the root, always." Ten is one past a multiple of nine, so the root advances by one and wraps round after nine. The wrapping is the part students drop.
• "A digital-root check proves a calculation is right." It proves the two sides agree on their position past a multiple of nine. A wrong answer can survive it. The chapter says the method was used to check calculations, and it is worth being exact about what such a check is worth.

Questions to check understanding

• Compute the digital root of a given number
• State the digital root of a number known to be a multiple of nine, and of one known to leave a stated remainder
• List which numbers in a stated hundred have a given digital root
• Describe the pattern of digital roots along a run of consecutive numbers, or along consecutive multiples of a given number
• Deduce the digital root of a number obtained by adding a fixed amount to a number whose root is known
• Find the digital root of an expression containing multiples of nine
• Solve a riddle whose clues are conditions on digits, digit count and digital root
• Use a digital-root check on an arithmetic calculation and say what it has and has not shown

Examples worth working on the board

Inputs. Values marked "printed" are the chapter's own.

• The definition and its worked instance (Part I, §5.2, p.130, printed subheading "Digital Roots"). Printed: take a number, add its digits repeatedly until one digit is left, and call that the digital root. The chapter's own example is 489710, with all three passes shown — the six digits totalling 29, then 29 giving 11, then 11 giving 2, and the root reported as 2. Use exactly this number; three passes is what makes the process feel like a process.
• The chapter's own prompt back to the nines test (same page): what property will the digital root have, given what was done for the divisibility shortcut for nine. This is the hinge of the whole topic and the chapter asks it without answering it.
• The open questions on Part I p.130, all left for the student:
  • Which numbers between 600 and 700 have digital root 5, which have 7, and which have 3.
  • Write the digital roots of any twelve consecutive numbers and say what you notice. Twelve is chosen so the cycle is seen more than once.
  • Take a run of multiples of 3 and work out each one's root; then do the same for a run of multiples of 4, and for a run of multiples of 6.
  • Find the digital roots of numbers that sit one past a multiple of 6, and say what is noticeable.
  • Explain the patterns noticed.
• The one statement the chapter does make (same page): the digital root of a multiple of nine is always nine. Note that this sits four pages after Part I p.125, which says that repeatedly summing digits gives the remainder on division by nine. Those two printed sentences pull against each other for multiples of nine, and section 5 exists to resolve the tension rather than let a student meet it unprepared.
• The riddle (Part I, §5.2, p.130, six lines of verse). Its conditions, given as data rather than as text: every digit is odd, and as small as an odd digit can be; the number has nothing in common with the root numbered one in the accompanying picture; and the count of its digits, the total of its digits and its digital root all point to the same value, which is the largest odd single digit. The answer is not printed anywhere in the chapter. Beside the verse is a cartoon shopfront with numbers on its awning, a badge reading as number one, a signboard naming a digital root club, and a smaller board reading members only. The picture is a joke about membership, not a source of data; read the lettering from the image if it is used.
• The historical note, printed (Part I, §5.2, p.131). The method of reaching a single digit by repeatedly adding digits is mentioned in the work Mahāsiddhānta by Aryabhata II, dated in the chapter to about 950 CE, and the chapter states that it was used to check arithmetic calculations.
• Exercise inputs, §5.2 "Figure it Out" (Part I p.131).
  • No. 1: an eight-digit number has digital root 5; what is the digital root of the number ten larger? Note that the digit count is stated and is irrelevant, which is part of the question.
  • No. 2: start from any number and build a sequence by adding eleven each time; describe the digital roots of that sequence.
  • No. 3: the digital root of the expression 9a + 36b + 13. The useful input is that the first two terms are multiples of nine whatever a and b are, which leaves only the third to think about.
  • No. 4: make conjectures about (i) how a number's parity relates to its digital root, and (ii) how its digital root relates to its remainder on division by three or by nine. Item (ii) is the topic's own thesis handed back as a conjecture to be discovered.
• The chapter prints no answers to any of these.

Figures to have open

• A collapsing-digits movement frame: the number's digits gathering into a total, that total's digits gathering again, and so on, with the discarded run-of-nines heap shown leaving at each pass. Standard schematic; it is the figure that makes the invariance visible and the chapter draws nothing here.
• A nine-position dial or ring, with the roots 1 to 9 around it, so that adding ten advances one step and the wrap is a physical motion rather than a rule. Standard schematic.
• A twelve-cell strip of consecutive numbers with their roots beneath, showing the repeat. Standard schematic.
• The chapter's shopfront cartoon on Part I p.130 is decorative and carries the riddle's joke rather than its data; it should not be reproduced, and the riddle works without it.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 5, "Number Play", §5.2 "Checking Divisibility Quickly", printed subheading "Digital Roots", Part I pp.130–131. The historical note and the "Figure it Out" items are on Part I p.131, immediately before §5.3 begins on the same page.
• Backward pointer inside the same chapter: the repeated digit-summing procedure and its meaning are introduced at Part I p.125, in the shortcut for nine, and the chapter tells the reader to recall it.
• Forward pointer inside the same chapter: the digit-reversal conjecture at Part I p.132 no. 4 rests on the digit total being unchanged by reordering.
• The chapter's SUMMARY, Part I p.134, does not mention digital roots; the material is treated as an extension of the nines shortcut rather than as a separate result.