The nine rule
Why do multiples of 9 add up to 9?
Real math, provable in one line
Because we count in tens, and 10 is one more than 9. Every 10, 100 or 1,000 is a pile of nines plus one, so adding up a number’s digits only throws away nines. A multiple of 9 stays a multiple of 9 all the way down, until one digit is left, and that digit is 9. The digits of 99 add to 18, not 9, but add again and you land on 9.
Keep adding the digits
Each place is a digit times a run of nines, plus the digit. The nines are all multiples of 9. Only the digits are left over.
The 9 times table, tap one
The proof
Why it works, in one line
Take 7,254. Written out by place, it is 7 × 1000 + 2 × 100 + 5 × 10 + 4. Now split every place value into nines plus one: 1000 is 999 + 1, 100 is 99 + 1, 10 is 9 + 1.
That turns the number into 7 × 999 + 2 × 99 + 5 × 9, which is a multiple of 9 no matter what the digits are, plus 7 + 2 + 5 + 4, which is just the digit sum.
So a number and the sum of its digits always differ by a multiple of 9. Whatever one leaves over after dividing by 9, the other leaves the same. If the number is a multiple of 9, its digit sum is too. Repeat until one digit is left, and a multiple of 9 can only end at 9. Mathematicians call that last digit the digital root.
Because 3 divides 9, the same split proves the rule for 3: a number is divisible by 3 exactly when its digit sum is. That is why multiples of 3 collapse only to 3, 6 or 9.
The finger trick
The 9 times table on ten fingers
Hold up ten fingers. To work out 9 times a number from 1 to 10, fold down that finger, counting from the left. The fingers to the left of the fold are the tens, the fingers to the right are the ones. For 9 × 7, fold the seventh finger: 6 on the left, 3 on the right, 63.
It works because 9 × n is 10 × (n − 1) + (10 − n). The fold splits your ten fingers into exactly those two numbers, and they always add to 9.
The list
What are the multiples of 9?
The multiples of 9 are the numbers you get by multiplying 9 by a whole number: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, and on forever, every ninth number.
There are 11 multiples of 9 up to 100, the last one 99. Up to 200 there are 22, the last one 198. Up to 1,000 there are 111, the last one 999.
How to tell if a number is a multiple of 9: add its digits. If the total is 9, 18, 27 or any other multiple of 9, the number is one too. If the total is still long, add its digits again. 4,788 gives 4 + 7 + 8 + 8 = 27, and 2 + 7 = 9, so 4,788 is a multiple of 9 (9 × 532).
History
Casting out nines
Using digit sums to check arithmetic is old enough to have a name, casting out nines. Add up the digits of each number in a sum, do the sum on those small numbers, and compare with the digits of your answer. If they disagree, you made a mistake. It cannot catch everything: swap two digits, writing 1324 for 1234, and the check still passes.
The Greek writers Hippolytus and Iamblichus described reducing numbers to their digit roots, but did not use it to check calculations. The earliest surviving work that does is the Mahāsiddhānta, written around 950 by the Indian mathematician and astronomer Aryabhata II. Around 1020 the Persian scholar Ibn Sina described it in full as the “Hindu method,” and Fibonacci set it out in his Liber Abaci. It reached Europe through Arabic writers, which is why it is sometimes called the Hindu check.
It stayed in schoolbooks for centuries. In 1824 a teenage Abraham Lincoln used casting out nines to check long multiplication in his cyphering book, a page of which survives at Columbia University.
Where vortex math comes in
Why 9 looks like it rules everything
Vortex math reads this rule as a sign that 9 is special in nature. Add 9 to any number and its digits still collapse to the same single digit. Multiply anything by 9 and it collapses to 9. Its proponents call 9 the control, or the singularity.
The rule is real, but it is a fact about the way we write numbers, not about 9. Count in base 12 and the number that behaves this way is 11, because 12 is one more than 11. Count in base 8 and it is 7. You can watch the pattern move in Is vortex math real?, and follow the other famous loop, 1 2 4 8 7 5.
Real vs legend
The nine rule, sorted
| Claim | Status |
|---|---|
| A number is divisible by 9 exactly when its digit sum is. | Real · Proven |
| The digits of every multiple of 9 add up to 9. | Keep adding · 99 gives 18 |
| Keep adding digits and every multiple of 9 ends at 9. | Real · Proven |
| The rule works for 3 as well. | Real · 3 divides 9 |
| It works the same in every number system. | No · base 12 uses 11 |
| It shows 9 has a special power in nature. | Legend |
Questions
Multiples of 9
Why do multiples of 9 add up to 9?
Because 10 is one more than 9. Every place value (10, 100, 1,000) is a run of nines plus one, so a number and the sum of its digits differ by a multiple of 9. A multiple of 9 therefore has a digit sum that is a multiple of 9, and if you keep adding digits you end at 9.
Do all multiples of 9 add up to 9?
Not in one step. 99 gives 9 + 9 = 18, and 9 × 111 = 999 gives 27. The digit sum is always a multiple of 9, though, so if you keep adding the digits of the answer, every multiple of 9 ends at 9.
What are the multiples of 9?
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, and so on, adding 9 each time. There are 11 up to 100 and 111 up to 1,000.
How can you tell if a number is a multiple of 9?
Add up its digits. If the total is a multiple of 9, so is the number. For example 4,788 gives 4 + 7 + 8 + 8 = 27, and 27 is 9 × 3, so 4,788 is a multiple of 9.
What is the trick for multiplying by 9?
Hold up ten fingers and fold down the one you are multiplying by, counting from the left. Fingers to the left of the fold give the tens and fingers to the right give the ones. For 9 × 4, fold the fourth finger: 3 left, 6 right, 36.
Does the digit rule work for 3?
Yes. A number is divisible by 3 exactly when its digit sum is, because 3 divides 9. That is why the digits of multiples of 3 always collapse to 3, 6 or 9.
Sources
Where this comes from
- Divisibility rule, the rule for 3 and 9 and its proof: Wikipedia.
- Digital root, definition and formula: Wikipedia. Eric W. Weisstein, “Digital Root,” MathWorld.
- Casting out nines, its limits and history: Wikipedia. Eric W. Weisstein, “Casting Out Nines,” MathWorld.
- Why the 9 times table hand trick works: Make Math Moments.
- “Introduction to Vortex Based Math,” fan site, 2010, on 9 as “the singularity”: archived copy.
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Poster
The nine rule and its one line proof, printed and framed.