# Digital Root The Digital Root is the sum of the individual digits of a number, repeating this process until we get a one-digit result.

### Example: 25

Sum the digits of 25: 2 + 5 = 7

Answer: the digital root of 25 is 7

### Example: 3456

Sum the digits of 3456: 3 + 4 + 5 + 6 = 18

But 18 has two digits, not one! So let's repeat the process.

Sum the digits of 18: 1 + 8 = 9

Answer: the Digital Root of 3456 is 9

We always end up with a number between 0 and 9.

## Useful

The digital root is useful as it helps us check the accuracy of our arithmetic, and can also check divisibility by 3 and 9.

Try adding the digital roots, their sum should also be correct:

### Example: 1234 plus 411

 Sum Digital Root 1234 1+2+3+4=10 → 1+0 = 1 1 + 411 4+1+1 = 6 + 6 1645 1+6+4+5=16 → 1+6 = 7 7

After doing the sum 1234+411=1645, check to see that the sum of the digital roots works out fine too. If it doesn't you must have made a mistake somewhere!

This also works when adding long lists of numbers.

### Check Subtraction

Same idea but using subtraction:

### Example: 1234 minus 411

 Sum Digital Root 1234 1+2+3+4=10, 1+0 = 1 1 − 411 4+1+1 = 6 − 6 823 8+2+3=13, 1+3 = 4 −5

Hang on ... minus 5? OK, when we get a negative digital root we can add 9

So −5 + 9 = 4, and it checks nicely.

Why did we add 9? Well the reason all this works is that 9 is the key number because it is one less than 10 (the basis of our decimal number system).

### Casting Out 9s

In fact the method we are using is sometimes called "casting out 9s": we are, in effect, casting out (getting rid of) all whole multiples of 9.

### Example: The digital root of 123

123 is also 1×100 + 2×10 + 3

Which is also 1×(99+1) + 2×(9+1) + 3×1

Which can be rearranged into 1×99 + 2×9 + 1×1 + 2×1 + 3×1

Ignoring all the multiples of 9 we get  1×1 + 2×1 + 3×1 = 1+2+3 = 6

### A Circle Explains It

It is like we don't care how many times we go around, just where we end up.

### Example: The digital root of 16

Here is a circle with 9 points (0,1,2,3,4,5,6,7,8): 16 goes once around, then ends up at 7: Or this way: 10 (which ends at 1) followed by 6, or just "1 + 6 = 7" Notice:

• 10 goes once around and ends at 1
• 20 goes twice around and ends at 2
• 30 goes three times around and ends at 3, etc

Likewise:

• 100 goes 11 times around and ends at 1
• 200 ends at 2
• 300 ends at 3

So we can just add up the digits to find where we end up!

Play with it here:

images/mod-anim.js

### Going Backwards

Remember earlier when we had  "1−6 = −5"  and we added 9 to get 4 ?

Well the circle explain how "−5" is the same as "4" : So the digital root is all about "where do we end up on the circle" and we ignore how many times we go around.

### Shortcuts

In fact there is a simple shortcut: find any digits that add up to 9 (one full rotation), and just ignore them!

### Example: The digital root of 453

4+5 is 9, so ignore them. So the digital root of 453 is simply 3

### Example: The digital root of 13331

Three 3's make 9, so ignore them. So the digital root of 13331 is 1+1=2

### Multiplication

We can also use digital roots to check Multiplication!

### Example: 345 times 107

 Sum Digital Root 345 Ignore 4+5: 3 × 107 1+7=8: × 8 36915 (Ignore 3+6 and 9) 1+5=6: 24

The digital root of 24 is 2+4=6, which matches the digital root of 36915. Yay!

### Division

Oh this is way too hard for me, you work it out!

### Decimals

And we can find the digital root of decimals, too:

1+2+5 = 8

### Example: 0.125 plus 0.125

0.125 + 0.125 = 0.25

Digital roots give us:

• 8 + 8 = 16, and 1+6 = 7,
• which matches the digital root of 0.25: 2+5 = 7

So we probably got that sum correct!

## Divisibility

Digital roots also help us check Divisibility (after dividing one number by another do we get a whole number answer) for both 3 and 9.

### Divisible by 9

Can we divide a number by 9 and get a whole number result? Only when the digital root is 9.

### Example: is 709 divisible by 9?

Sum the digits: 7+0+9 = 16, repeat for 16: 1 + 6 = 7

Answer: NO, 709 is not divisible by 9

(Check: 709/9 = 7879, not a whole number)

### Example: is 5436 divisible by 9?

Sum the digits: 5+4+3+6 = 18, repeat for 18: 1 + 8 = 9

Answer: YES, 5436  is divisible by 9

(Check: 5436/9 = 604 exactly)

### Divisible by 3

Can we divide a number by 3 and get a whole number result? Only when the digital root is a multiple of 3.

### Example: is 428 divisible by 3?

Sum the digits: 4+2+8 = 14, repeat for 14: 1 + 4 = 5

Answer: NO, 428 is not divisible by 3

(Check: 428/3 = 14223, not a whole number)

### Example: is 1725 divisible by 3?

Sum the digits: 1+7+2+5 = 15, repeat for 15: 1 + 5 = 6

Answer: 6 is a multiple of 3, so YES, 1725 is divisible by 3

(Check: 1725/3 = 575 exactly) 