Cambridge O Level Computer Science · Syllabus 2210 · Data Representation
Denary
What is Denary?
Denary is the base 10 number system used by people, with the ten digits 0 to 9 and column values that are powers of ten.
This definition is part of the Data Representation chapter in Cambridge O Level Computer Science.
Denary in context
Binary addition works exactly like denary column addition, but you carry when a column reaches two instead of ten. Align the two numbers so their least significant bits are in the same column, add each column from the right, and carry a 1 into the next column whenever the column total is 2 or 3.
Common mistakes with Denary
- M8. "The most significant bit is simply a minus sign." Why it fails In two's complement the leading 1 is a place value of −128 that is added to the rest — it is not a symbol you can detach and ignore. Correct model Read 11010011 as −128 + 64 + 16 + 2 + 1 = −45, not as "minus 1010011" = −83. Exam-safe "In two's complement the most significant bit has a place value of −128; a leading 1 indicates a negative number but the other bits still add positive values to it." Test yourselfHideQ. What denary value is the two's-complement pattern 11111111?A. −1, not −127.
Examiner tips on Denary
- The shortcut worth knowing. There is no direct route between denary and hexadecimal that is faster than going through binary for small values, but for anything over one byte the repeated-division method is quicker. And hexadecimal and binary convert directly, four bits at a time, with no arithmetic at all — so if you are ever asked for denary to hexadecimal and get stuck, convert to binary first, then group in fours.
- The seven things to state in a shift answer the original bit pattern; the direction of the shift; the number of places; which bits were discarded; the final 8-bit pattern (all eight bits, zeros included); the denary result; whether the simple ×2 or ÷2 interpretation still holds. Points 4 and 7 are where most marks are lost. Never answer a shift question without checking what fell off the end.
Questions students ask about Denary
What is the difference between binary, denary and hexadecimal?
Denary is base 10, the number system people normally use, with column values that are powers of ten. Binary is base 2, using only the digits 0 and 1 with column values that are powers of two — the only form a computer actually stores. Hexadecimal is base 16, using the digits 0 to 9 and the letters A to F, with column values that are powers of sixteen. Each hexadecimal digit represents exactly four binary bits, so it is a much shorter way for people to write the same bit pattern.