# Representing numbers in different bases a relationship

### Binary Numbers and ASCII – diveintocs In mathematical numeral systems, the radix or base is the number of unique digits, including the number one hundred, while ()2 (in the binary system with base 2) represents the number four. 16, Hexadecimal system, Often used in computing as a more compact representation of binary (1 hex digit per 4 bits). For example, to represent the number in base 10, we know we place a 3 in the . the base 2 factors to memory as an aid to quickly correlate the relationship . Positional notation or place-value notation is a method of representing or encoding numbers. In a tablet unearthed at Kish (dating from about BC), the scribe . So binary numbers are "base-2"; octal numbers are "base-8"; decimal.

Any inadvertent covering of a hole will cause an incorrect value to be read, causing undefined behaviour. Parity allows a simple check of the bits of a byte to ensure they were read correctly. We can implement either odd or even parity by using the extra bit as a parity bit. In odd parity, if the number of 1's in the 7 bits of information is odd, the parity bit is set, otherwise it is not set. Even parity is the opposite; if the number of 1's is even the parity bit is set to 1. In this way, the flipping of one bit will case a parity error, which can be detected. XXX more about error correcting 16, 32 and 64 bit computers Numbers do not fit into bytes; hopefully your bank balance in dollars will need more range than can fit into one byte!

Modern architectures are at least 32 bit computers. This means they work with 4 bytes at a time when processing and reading or writing to memory. We refer to 4 bytes as a word; this is analogous to language where letters bits make up words in a sentence, except in computing every word has the same size! The size of a C int variable is 32 bits.

Modern architectures are 64 bits, which doubles the size the processor works with to 8 bytes. Kilo, Mega and Giga Bytes Computers deal with a lot of bytes; that's what makes them so powerful! We need a way to talk about large numbers of bytes, and a natural way is to use the "International System of Units" SI prefixes as used in most other scientific areas.

So for example, kilo refers to or units, as in a kilogram has grams.

### Radix - Wikipedia

However, or is a round number — — and happens to be quite close to the base 10 meaning value of "kilo" as opposed to Thus bytes naturally became known as a kilobyte. The next SI unit is "mega" for and the prefixes continue upwards by corresponding to the usual grouping of three digits when writing large numbers. Other bases have been used in the past, and some continue to be used today. For example, the Babylonian numeral systemcredited as the first positional numeral system, was basebut it lacked a real 0 value.

Zero was indicated by a space between sexagesimal numerals.

## Chapter 2. Binary and Number Representation

Nor was it used at the end of a number. Only context could differentiate them. The polymath Archimedes ca. With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables as does positional notation and could produce practical results quickly.

For four centuries from the 13th to the 16th there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus.

Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries. In a similar manner, we can specify numbers in other "bases" besides 10using different digits that correspond to the coefficients on the powers of the given base that must be added together to obtain the value of our number.

This is consistent with base 10 numbers, where we use digits For smaller bases, we use a subset of these digits. For example, in base 5, we only use digits ; in base 2 which is also called binarywe only use the digits 0 and 1.

For larger bases, we need to have single digits for values past 9. Hexadecimal base 16 numbers provide an example of how this can be done. In this way, we have digits corresponding towhich is what we need. In these instances, the context of their use usually makes the base clear.