IGCSE Computer Science 0478 — 1.1 Number Systems Hangman

The 23 key terms

1binary

A number system using only 0s and 1s (bits). Digital systems use binary, as on/off signals, to represent all data: text, images, sound and video. 8 bits make 1 byte.

2bit

A bit is a single binary digit that is either 0 or 1, like a switch that is off (0) or on (1). A group of 8 bits is called a byte.

3number base

How many different digits a number system uses: denary is base 10 (digits 0-9), binary is base 2 (digits 0-1) and hexadecimal is base 16 (digits 0-9 and A-F).

4denary place value

Each denary column is worth 10 times the column to its right (1, 10, 100, 1000). A number is the sum of each digit times its column value, e.g. 345 = 300 + 40 + 5.

6Add the place values that have a 1

To read a binary number, add up the place values (1, 2, 4, 8...) that have a 1 under them and ignore the 0s. For example, 1011 is 8 + 2 + 1 = 11.

78-bit binary: 128 64 32 16 8 4 2 1

A byte has 8 bits with place values 128 down to 1. Add the place values that are 1s. The smallest number is 0 and the largest is 11111111 = 255, so a byte stores 0 to 255.

8subtract place values

To write a number in 8-bit binary, go from 128 down to 1. If the place value fits in what is left, write 1 and subtract it. If not, write 0. Check by adding the place values.

11add the place values with a 1

To convert 16-bit binary to denary, write the place values (32768 down to 1) over the bits, skip the 0 columns, then add the values that have a 1 under them. Leading zeros add nothing.

12denary to 16-bit binary

Work from 32768 down to 1. Write 1 and subtract if the place value fits in the remainder, otherwise write 0. Keep leading zeros so the answer has 16 digits.

130123456789ABCDEF

The 16 hex digits: 0-9, then A=10, B=11, C=12, D=13, E=14, F=15. Each digit stands for one 4-bit value, and F (15) is the largest single hex digit.

15hex to denary

Multiply each hex digit's value (A-F = 10-15) by its column value (1, 16, 256, 4096), then add the results. Example: 2F = 2 x 16 + 15 = 47.

16divide by 16

Divide the denary number by 16 again and again. Read the remainders from last to first, writing 10 to 15 as A to F. Check by converting the hex back to denary.

17binary to hex

Split the bits into groups of four from the right (nibbles), pad the left group with zeros if needed, then replace each nibble with its hex digit.

19hexadecimal

A base-16 system that shortens binary. One hex digit stands for four bits, so values are shorter to read, easier to copy without mistakes and quick to convert to and from binary.

22binary addition

Add two 8-bit binary numbers column by column from the right. Carry any 1 to the next column on the left. 1 + 1 = 10, so write 0 and carry 1. Check by adding the denary values.

23overflow

An error when an 8-bit addition gives a result over 255. It needs a 9th bit, so the carry out of the leftmost column is lost and the stored result is wrong.

24logical left shift

Moves every bit one place left. The leftmost bit falls off and is lost, and a 0 fills the empty place on the right. In 8 bits, 00000101 becomes 00001010.

25logical right shift

Moves every bit one place right. The bit at the right end is lost and a 0 fills the empty left end. For a positive integer this halves the value (rounded down).

26shift n places

Shifting by n places moves every bit n positions, losing n bits at one end and adding n zeros at the other. Shift all 8 bits out and you get 00000000.

27n shifts = 2^n

Each left shift multiplies a binary number by 2 and each right shift divides it by 2, so n shifts multiply or divide by 2^n. Bits shifted off the end are lost.

28two's complement

A way to store negative integers in binary. The leftmost bit shows the sign: 0 means positive, 1 means negative. Positive numbers look the same as ordinary binary.

32positive two's complement

A denary number from 0 to 127 is written as ordinary binary, padded with zeros to 8 bits so the leftmost sign bit is 0. Numbers above 127 cannot be stored.

33flip the bits then add 1

Write the positive value in 8-bit binary, flip every bit, then add 1. Example: 5 is 00000101, flip to 11111010, add 1 to get 11111011 (-5).