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Chapter 2

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Mathematics and Statistics
MATH 1013
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Chapter 2: Binary Values and Number Systems 2.1 Numbers and Computing • Number a unit of an abstract mathematical systems subject to the laws of arithmetic • Natural Number the number 0 and any number found by repeatedly adding 1 to 0 • Integer any natural or negative number (whole numbers; no decimals) • Rational Number an integer or quotient of two integers 2.2 Positional Notation • Base the foundational value of a number system, which dictates the number of digits and the value of digit positions (decimal system uses base 10) • Positional notation a system of expressing numbers in which the digits are arranged in succession, the position of each digit has a place value, and the number is equal to the sum of the products of each digit by its place value • Binary, octal and hexadecimal o Binary uses base 2, octal base 8, hexadecimal base 16 o To represent values greater than 9, we use letters (A-F) o Numbers in a system can never be greater than the base number  Eg. Octal uses 0-7, never anything higher o To convert other bases to base 10, use methods discussed in class o 0 in the rightmost position and 1 in the second position represents the value of the base itself (i.e 10 in octal is 8 in decimal, 10 in binary is 2 in decimal) • Arithmetic in other bases o Refer to class notes CSE_B_arithmetic • Power of 2 number systems o Refer to class notes CSE_B_arithmetic • Converting from base 10 to other bases o Refer to class notes CSE_B_arithmetic o Divide the decimal number by the new base, make the remainder the next digit in the answer, replace the decimal number with the quotient • Binary values and computers o Computers today are binary machines o They can only use low voltage signals or high voltage signals (0 and 1 respectively) o Each storage location must contain either 0 or 1 o each storage unit is called a binary digit or a bit o 8 bits create a byte, bytes created words when grouped together o Number of bits in a word is the word length of the computer (32 or 64 bit) Chapter 3: Data Representation 3.1 Data and Computers • Data basic values or facts • Information data that has been organized or processed in a useful manner • Computers are now multimedia devices o They deal with different data such as numbers, text, audio, images, video • All data on a computer is represented in binary as 0s and 1s • Data compression reducing the amount of space needed to store a piece of data • Bandwidth the number of bits or bytes that can be transmitted from one place to another in a fixed amount of time • Compression ratio the size of the compressed data divided by the size of the uncompressed data (closer the number is to 0, the tighter the compression) • Lossless compression a data compression technique in which there is no loss of information • Lossy compression a data compression technique in which there is loss of information • Analog and Digital Data o Computers are finite devices; they only have so much room to store and manipulate a certain amount of data o Analog dataa continuous representation of data, analogous to the actual information it represents (ie a thermometer) o Digital data a discrete representation of data, breaking the information up into separate elements o Analog data is directly proportional to the continuous, infinite world around us, therefore computers cannot work well with analog data o Digitize the act of breaking information down into discrete pieces o The discrete elements are then individually represented using binary digits o The reason to use binary systems as opposed to other number systems (ie decimal) is because devices that store and manage binary data are less expensive and more reliable if they only have to represent one of two possible values o An analog signal continually fluctuates up and down in voltage. But a digital signal has only a high or low state, corresponding to the two binary digits. o All electronic signals (both analog and digital) degrade as they move down a line. That is, the voltage of the signal fluctuates due to environmental effects. o Analog signals lose information when they degrade o Digital signals have a threshold where values above the threshold are considered high voltage and values below are low voltage o Periodically, digital signals are reclocked to regain its original shape, preventing loss of data • Binary representation o n bits can represented 2 things because 2 combinations of 0 and 1 can be made from n bits o ex. 4 bits represents 16 things, 5 bits 32 o 2 bits represent 4 things ex. 00 01 10 11 3.2 Representing Numerical Data Representing negative values • Signed-magnitude representation o Number representation in which the sign represents the ordering of the number (negative or positive) and the value represents the magnitude o Uses the number line; positive is to the right of 0 and negative is to the left o This creates a problem when using 0 (plus 0 or minus 0)  We don’t consider negative 0, only positive 0  The two representations of 0 can cause unnecessary complexity with computers, therefore they use other representations of numbers • Fixed-sized numbers (or complementary/tens complement) in prof notes) o If we allow a fixed number of values, we can represent numbers as integer values where half represent positive an half represent negative o The sign is determined by the magnitude of the number  Ex.0 to 49 are positive, 50 to 99 are negative (-50 to -1) o
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