Align equal place values, exchange two lower-place units for one higher-place unit, and reverse that exchange when a subtraction column needs a borrow.
Addition bundles two units and carries one place left. Subtraction unbundles one higher-place unit into two lower-place units.
Binary digits must be aligned by place value. Added leading zeros are placeholders; they do not change a numeral's value.
Align the ones columns on the right.
Divide the column total by 2: its remainder is the written bit.
Include the incoming carry before finding the new bit.
Pad the shorter addend with leading zeros.
A carry can create a new total of two in the next column. This is a carry chain.
Columns that send a carry: 4
Longest consecutive carry chain: 4
Count a column only if it sends a carry to its left. Do not count the final new column that merely receives the last carry. These questions use the printed numerals, independent of the explorer settings.
A carry continues whenever it meets another 1.
If a zero column must pass a borrow through, it borrows from farther left and then passes one unit right.
For the last two questions, the current column is 0−1 with no borrow already owed. State the written bit and the new borrow flag separately.
One higher-place unit is worth two units here.
The row labeled “borrow in flags” records whether each column owes one unit to the column on its right. These are borrow flags. Subtract that owed unit before subtracting the bottom bit. A zero column may need to borrow from farther left to pay what it owes.
A zero column that owes a borrow must borrow again from the left.
Give each subtraction result in binary. The final choice compares numerical values, not the lengths of the written numerals.
Subtract the bottom bit and the borrow owed to the right.
All five answers are binary. In the final expression, do the addition first and then the subtraction. Within each column calculation, work from the rightmost place toward the left.
Align equal places, then check by converting the inputs and result to decimal.
Align equal places, then check by converting the inputs and result to decimal.
Align equal places, then check by converting the inputs and result to decimal.
Align equal places, then check by converting the inputs and result to decimal.
Align equal places, then check by converting the inputs and result to decimal.
Correct all eight questions to complete the workshop. Write calculation results in binary; write counts and requested decimal values in decimal.
Answers: 10001₂; 100100₂; 1111₂; 11111₂; 5; 35; 11000₂; 2. Decimal checks: 11+6=17; 29+7=36; 16−1=15; 42−11=31; 37−13=24. Five consecutive 1s each send a carry when adding 1.
Answers: 100011₂; 1001₂; write 1/carry 1; 1010101₂; 32. Decimal checks are 26+9=35, 12−3=9, 106−21=85, and 22+10=32.
This certifies that
aligned powers of two, completed binary additions and subtractions, explained carry and borrow chains, and verified results in decimal.