Carry flag
In computer processors the carry flag (usually indicated as the C flag) is a single bit in a system status register/flag register used to indicate when an arithmetic carry or borrow has been generated out of the most significant arithmetic logic unit (ALU) bit position. The carry flag enables numbers larger than a single ALU width to be added/subtracted by carrying (adding) a binary digit from a partial addition/subtraction to the least significant bit position of a more significant word. This is typically programmed by the user of the processor on the assembly or machine code level, but can also happen internally in certain processors, via digital logic or microcode, where some processors have wider registers and arithmetic instructions than (combinatorial, or "physical") ALU. It is also used to extend bit shifts and rotates in a similar manner on many processors (sometimes done via a dedicated flag). For subtractive operations, two (opposite) conventions are employed as most machines set the carry flag on borrow while some machines (such as the 6502 and the PIC) instead reset the carry flag on borrow (and vice versa).

Uses 
The carry flag is affected by the result of most arithmetic (and typically several bit wise) instructions and is also used as an input to many of them. Several of these instructions have two forms which either read or ignore the carry. In assembly languages these instructions are represented by mnemonics such as ADD/SUB, ADC/SBC (ADD/SUB including carry), SHL/SHR (bit shifts), ROL/ROR (bit rotates), RCR/RCL (rotate through carry), and so on. The use of the carry flag in this manner enables multi-word add, subtract, shift, and rotate operations.

An example is what happens if one were to add 255 and 255 using 8-bit registers. The result should be 510 which is the 9-bit value 111111110 in binary. The 8 least significant bits always stored in the register would be 11111110 binary (254 decimal) but since there is carry out of bit 7 (the eight bit), the carry is set, indicating that the result needs 9 bits. The valid 9-bit result is the concatenation of the carry flag with the result.

For x86 ALU size of 8 bits, an 8-bit two's complement interpretation, the addition operation 11111111 + 11111111 results in 111111110, Carry_Flag set, Sign_Flag set, and Overflow_Flag clear.

If 11111111 represents two's complement signed integer −1 (ADD al,-1), then the interpretation of the result is -2 because Overflow_Flag is clear, and Carry_Flag is ignored. The sign of the result is negative, because Sign_Flag is set. 11111110 is the two's complement form of signed integer −2.

If 11111111 represents unsigned integer binary number 255 (ADD al,255), then the interpretation of the result would be 254, which is not correct, because the most significant bit of the result went into the Carry_Flag, which therefore cannot be ignored. The Overflow_Flag and the Sign_Flag are ignored.

Another example may be an 8-bit register with the bit pattern 01010101 and the carry flag set; if we execute a rotate left through carry instruction, the result would be 10101011 with the carry flag cleared because the most significant bit (bit 7) was rotated into the carry while the carry was rotated into the least significant bit (bit 0).

The early microprocessors Intel 4004 and Intel 8008 had specific instructions to set as well as reset the carry flag explicitly. However, the later Intel 8080 (and Z80) did not include an explicit reset carry opcode as this could be done equally fast via one of the bitwise AND, OR or XOR instructions (which do not use the carry flag).

The carry flag is also often used following comparison instructions, which are typically implemented by subtractive operations, to allow a decision to be made about which of the two compared values is lower than (or greater or equal to) the other. Branch instructions which examine the carry flag are often represented by mnemonics such as BCC and BCS to branch if the carry is clear, or branch if the carry is set respectively. When used in this way the carry flag provides a mechanism for comparing the values as unsigned integers. This is in contrast to the overflow flag which provides a mechanism for comparing the values as signed integer values.

Vs. borrow flag 

While the carry flag is well-defined for addition, there are two ways in common use to use the carry flag for subtraction operations.

The first uses the bit as a borrow flag, setting it if a<b when computing a−b, and a borrow must be performed. If a≥b, the bit is cleared. A subtract with borrow (SBB) instruction will compute a−b−C = a−(b+C), while a subtract without borrow (SUB) acts as if the borrow bit were clear. The 8080, 6800, Z80, 8051, x86 and 68k families (among others) use a borrow bit.

The second uses the identity that −x = (not x)+1 directly (i.e. without storing the carry bit inverted) and computes a−b as a+(not b)+1. The carry flag is set according to this addition, and subtract with carry computes a+not(b)+C, while subtract without carry acts as if the carry bit were set. The result is that the carry bit is set if a≥b, and clear if a<b. The System/360, 6502, MSP430, COP8, ARM and PowerPC processors use this convention. The 6502 is a particularly well-known example because it does not have a subtract without carry operation, so programmers must ensure that the carry flag is set before every subtract operation where a borrow is not required.

Most commonly, the first alternative is referred to as a "subtract with borrow", while the second is called a "subtract with carry". However, there are exceptions in both directions; the VAX, NS320xx, and Atmel AVR architectures use the borrow bit convention, but call their a−b−C operation "subtract with carry" (SBWC, SUBC and SBC).  The PA-RISC and PICmicro architectures use the carry bit convention, but call their a+not(b)+C operation "subtract with borrow" (SUBB and SUBWFB).

The ST6 8-bit microcontrollers are perhaps the most confusing of all. Although they do not have any sort of "subtract with carry" instruction, they do have a carry bit which is set by a subtract instruction, and the convention depends on the processor model. The ST60 processor uses the "carry" convention, while the ST62 and ST63 processors use the "borrow" convention.

See also
 Binary arithmetic
 Half-carry flag
 Status register

References

External links
 Carry Flag and Overflow Flag in binary arithmetic
 Carry Bit: How does it work?

Computer arithmetic