If we take a three-digit number – for example 348 – and we multiply it by 1001we get 348,348, that is, exactly the starting number repeated twice! And this happens for any three-digit number. You can try!
But how come this happens mathematical magic? In truth the explanation is very simple and becomes very clear if we apply the distributive property of multiplication. Multiplying by 1001, in fact, is equivalent to multiplying by (1000 + 1). So let’s see what happens if we multiply a three-digit number, like 348, by 1001:
348 × 1001
= 348 × (1000 + 1)
= 348,000 + 348
= 348,348
In short, by multiplying by 1000 we add three zeros to the starting number obtaining 348,000, and then adding 348 × 1 = 348 you get just the starting number doubled: 348,348.
This little game works with three-digit numbers precisely because multiplying by 1000 gives three zeros after the starting number.
And what if we try to do the same with one or two digit numbers? The magic still works, but with the addition of a few zeros.
If we try to do this, for example, with the number 2, we get:
2 × 1001
= 2000 + 2
= 2002
The number in this case repeats but with two zeros in the middle. If we do the same thing with a two-digit number – for example 21 – the number of zeros in between is reduced to 1:
21 × 1001
= 21000 + 21
= 21021
If, however, we increase the number of digits, that is, if the digits of the number are greater than three, the game “breaks”. Let’s see why.
If we consider for example the number 1221 and we multiply it by 1001 we get
1221 × 1001
= 1,221,000 + 1221
= 1222221
This is because the number of digits to be added exceeds the number of zeros in the first addend.
