# why sum of digits $(a+b)$ = sum of digits of $(a)$ + sum of digits of $(b)$

eg:

Sum of digits $(12+14)$ = sum of digits $(26)$ = $8$

sum of digits of $(12)$ = $3$ sum of digits of $(14)$ = $5$ adding these gives $8$

How this can be proved mathematically?

Will this be true for all integers (positive and negative)?

EDIT: We need to repeatedly find sum of digits as given below

  Sum of digits (9+1)=Sum of digits (10)=1

sum of digits (9)=9,
sum of digits (1)=1

adding gives 10, again, sum of digits of 10 gives 1,

• This is wrong! Try 9 + 1! – Tzimmo Oct 7 '15 at 13:47
• You probably mean digit sum which is different from sum of digits. – Shailesh Oct 7 '15 at 13:48
• l actually meant what @Atrthur mentioned. for eg, sum of digits of (9+1)=10=1 ; 1+0 gives same – Kiran Oct 7 '15 at 13:49
• Do you mean digital root? – lhf Oct 7 '15 at 13:52
• yes, i meant digital root, i was not aware of this before. Now got to know thx – Kiran Oct 7 '15 at 13:56

Perhaps you meant to refer to the digital root instead?

Call the digital root of $x$ as the function $dr(x)$.

Then $$dr(a + b) \equiv \left(dr(a) + dr(b)\right) \pmod 9$$ so that your claim follows.

• yes, i didnot know this term. I meant the same and this property was noticed by me while my studies. I will learn it now. thx – Kiran Oct 7 '15 at 13:57
• so for digital root, is this true always? – Kiran Oct 7 '15 at 13:59

Counterexample:

sum of digits (9)=9

sum of digits (1)=1

sum of digits (9+1)=(10)=1

$9+1\neq 1$

Your algorithm only works if you don't have carry-overs...

• Sorry, if i was not clear. What I meant is, sum of digits (9)=9, sum of digits (1)=1, adding gives 10, again, sum of digits of 10 gives 1. – Kiran Oct 7 '15 at 13:51

Sum of digits of $999999999$ is $81$.

Sum of digits of $1$ is $1$.

Sum of digits of the sum $1000000000$ is $1$.

$81+1 = 82$. Sum of its digits is $10$. Sum of its digits is $1$.

Is that what you mean? Keep doing it until you have a single digit?

• yes, this is exactly i meant. thx. through one of the answer given, i see this is digital root, right? – Kiran Oct 7 '15 at 13:59

@kiran this is just property of numbers. Digital sum & digital roots are the same & a/C to your question I think you are asking for digital root. So friend this is powerful technique invented by our ancient great mathematicians.

• thx a lot. i was not knowing these terms. Now it is clear. – Kiran Oct 7 '15 at 14:02
• Bro one new name T9 method. This also called to this method – Vikas.Ghode Oct 7 '15 at 14:04