In binary arithmetic 1 + 1 = 10 ? Why is this?

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Twinkle twinkle little star, tell me how many stars?

Binary says there are 11 stars.

Decimal says there are 3 stars.

The number of stars did not change, the number system changed.

We have 10 fingers so, we chose decimal number system.

Computers have 2 states (OFF & ON) so, they chose binary number system.

What is a number system?

A number system is a nomenclature of counting. Binary number system has two face values, 0 and 1. These face values are permuted in the digits in the following manner:

Binary

0

1

both face values permuted in one digit so go to next higher digit , represented by

10

11 go to three digit

100

keeping second digit constant and permuting first digit

101

110

111

all permutations of three digit number over so, go to four digit

1000

1001

1010

1011

1100

1101

1110

1111

And so on.

Similarly in decimal :

There are ten face values to be permuted. After 9 comes two digit 10, keeping second digit 1 constant, first digit is permuted. After 19, the permutation of second digit occurs which is two so, 20 and so on. When we reach 99, all permutations of two digit number are over so reach three digit and so on.

Now coming to your question,

Why 1+1=10 in binary?

It means go one step forward from 1. What comes after 1? By referring above, we find that it is 10. So it is equivalent to 9+1 in decimal.

Why 10–1 = 1 in binary?

It means go one step backward from 10. By referring above, we find that it is 1. It is equivalent to 10–1=9 in decimal.

1 & 9 are the last face values of binary and decimal systems, respectively.