The sum of 111 +10 = 1001. I don’t quite understand how that answer was calculated. I got 101 as an answer , because the answer is 121 and wherever there was an even number I put 0 and odd number I put 1, so 121 became 101. I don’t understand why there is an extra 0 .

Side note: I understand how to convert a decimal to binary , so anyway to build on that to solve answer ?

Relevance
• When you learned to add decimal numbers, you had to "carry" numbers bigger than 10.

e.g. 271 + 154 = 3(twelve)5 -> 425

In binary you have to carry 2's.

2 doesn't become 0, it becomes 10.

121 -> 201 -> 1001

• Binary is based on having only two digit values: 0 and 1

so, let's look first at the numbers in order from smallest (0) to larger (in other words, just adding 1 to the lower number and going up to, say, 10000

The decimal equivalent is where the ==> points to.

But the important part is to watch the pattern of the binary numbers

0 + 0 = 0 ==>  0

0 + 1 = 1 ==>  1

1+ 1 = 10 ==> 2

10 + 1 = 11 ==>  3

11 + 1 = 100 ==>  4

100 + 1 = 101 ==> 5

101 + 1 = 110 ==> 6

110 + 1 = 111 ==> 7

111 + 1 = 1000 ==> 8

1000 + 1 = 1001 ==> 9

1001 + 1 = 1010 ==> 10

1010 + 1 = 1011 ==> 11

1011 + 1 = 1100 ==> 12

1100 + 1 = 1101 ==> 13

1101 + 1 = 1110 ==> 14

1110 + 1 = 1111 ==> 15

1111 + 1 = 10000 ==> 16

111 is decimal 7

10 is decimal 2

7 + 2 = 9

so 111 + 10 = 1001

or...

1 + 0 (for the rightmost digit)  = 1

for the next digit

1 + 1 = 10, so the next shown digit is 0, with a carry of 1

1 + 1 (the carry digit) = 10 so next shown digit is 0 with a carry of 1

0 + 1 (carry digit) = 1 so the leftmost digit is 1

to get 1001

Hope this helps --

• Remember the rules of adding binary. We use 0 and 1 which the computer can only understand, so adding binary is different from that math you know.

1 + 0 = 1

1 + 1 = 10

0 + 1 = 1

1 + 1 = 10  you carry 1 from the 3rd digit on the top of 1

then add 1 + 1 = 10. Your result should be 1001..

1

111

+10

------

1001

• Actually it’s somewhat like regular addition: Let’s do it:

1 1 1    Add the first column, that’s 1, so bring it down to the answer.

10        Add the second column. That’s actually 2+2 =4. So that has to be carried     ------                to         the   4  column, the next column to the left.

1001 Add the carry to the next column. That’s two fours = 8. So that has to be moved            to the next Column which is an eight.

The idea is that if you add two ones, you have to put down a zero and add one to the next column.

1*8 +0*4 +0*2 +1*1 = 9 in decimal

• 7 + 2 = 9 (decimal)

111 + 10 = 1001 (Binary)

• in binary the only legal numbers are 0 and1

after a one is a carry  of one  to the next  higher bit

so

111

010

===

.....1

.10 with the 1 carried

1001

that  is a simple way to do it

for this type  of  typing text

the more proper hand way would be to  put a 1 over the  next bit higher

• you do know that 121 in base 10 is 100 + 20 + 1...thus you have 1 (10² ) , 2 (10^1) , & 1 (10^0 ).....so 121 does not exist in base 2 as that would say you have  1 ( 2²) , 2 (2^1) ≡ 4 = 2² (which means you have 2 (2²) ) , and 1 (2^0 )....ie. no coefficient 2 in the base 2 system......in base 2 system the sum 111 + 10 means { in base 10  (4 + 2 + 1 ) + ( 2 + 0) = 4 + 4 + 1 = 8 + 2 = 2^3 + 2^0 ≡ base 2 ( 1 0 0 1)

•   111

+010

Starting with the first digit on the right.   0 + 1 = 1

Go to the middle digit  1 + 1 = 4   Since 4 is a place value you don't have any 2's so the binary digit is 0 BUT you have to carry the place value

Last column on the left. 1 + 1 = 0 because you have nothing in the 4's

place value. But you carry the place value.  In other words you have 4 + 4 = 8

One helpful hint to add and subtract binary numbers, convert them to decimal numbers first because it will make it easier to understand the carrying of place values.

Subtraction in binary is quite interesting because of the borrowing.

Then there is multiplication and division in binary

• In Binary ,when adding 1 + 1  it = '10'

In decimal ,when adding 1 + 1 = 2  (The usual addition )

So in binary

100 + 11 = 111

but

101 + 11 = 1000

In the units column we have 1 + 1 = 10

So 'o' in the units column , carry '1' in to the 'tens' column.

In the 'tens' column we have 1 + 1(carried forward) = 10

So '0' in the 'tens' column , carry '1' in to the 'hundreds' column.

In the 'hundreds' column we have '1 + 1(carried forward) = 10

So '0' in the hundreds column , carry '1' in to the 'thousands' column.

Since there are no other numbers in the 'thousands'column we write '1' .

NB The binary system is  used in computers because electrical charge, on microchip boards',  is either 'on' or 'off' , or the electrical charge is either '+' or '-'

• Its just like normal addition. When it goes overboard, you have a "carry".

Lets write 10 as 010 for visual simplicity. [like if you had the number 362, you can write it as 0362 or 00362 etc].

0111 +

0010  becomes

1

111

010

-----

The right column is 1+0 = 1

The center column is 1+1 = 10. We went overboard. You have a (1)  to carry.

The left column becomes (1)+1+0 = 10

So you end up with (here again, for visual simplicity, I put an extra '0' to the left of each number)

01

0111

0010

----

1001

This carry "pattern" is *exactly* as in normal base ten addition. The carry goes to the next left column. Note that you said that your answer was 101. But that's logically impossible since you are adding 111 & something. So the answer will be bigger than 111. 101 is not bigger than 111. Only numbers 1000 and beyond are bigger than 111.Also, you said you got 121. That too is impossible. In binary, there are no "2". A 2 is NOT a 0. The decimal 2 is 10 in binary.