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Topic: Signed number representations

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	Number Systems
		Sign magnitude: the most significant bit is assigned to the algebraic sign.
		It is a natural kind of representation of signed integers, but is not used much because it is awkward for doing arithmetic compared to a system like 2's complement.
		One logical way to represent signed integers is to have enough range in binary numbers so that the zero can be offset to the middle of the range of positive binary numbers.
	hyperphysics.phy-astr.gsu.edu /hbase/electronic/number2.html (330 words)

	Signed Integer Coding (Site not responding. Last check: 2007-10-03)
		-bit representations are the same for representable non-negative integers.
		At times, it is necessary to coerce a signed integer from one type to another, resulting in a change in the number of bits used to represent the number.
		When a signed integer is coerced from a smaller length to a larger length the original bits are padded on the left with copies of the sign bit of the original number.
	www.d.umn.edu /~gshute/cs3011/asm/signed.html (882 words)

	Number Representations
		Since sixteen distinct symbols are needed to represent numbers in hexadecimal (in general b symbols are needed to represent numbers base b), the usual digits 0 through 9 are used and letters A through F are used to represent 10 through 15, respectively.
		It is customary, rather than use the subscript 16 to indicate a hexadecimal number, to follow the hex number with an H (or h) to indicate that it is in hexadecimal form.
		Adding and subtracting whole numbers (the only operations and types of numbers we will consider here) in binary is very similar to performing those same operations in decimal--there's the usual "carrying" (for addition) and "borrowing" (for subtraction) that we learned early on.
	www.warren-wilson.edu /~teller/courses/mat201/lang.htm (1741 words)

	CmpSci 535 Lecture 6
		With binary numbers we do not have the convenience of such a notation, although we could declare one bit (say bit 31) to be a sign bit and the rest of the number to be the quantity.
		Addition involves aligning the mantissas of the two numbers (determining the difference between their exponents and then shifting the smaller of the two numbers right by that many positions, with the implicit one made explicit), then adding the mantissas and if there is an overflow renormalizing by shifting the result one to the right.
		In single-precision numbers there is a one-bit sign, 23 bits in the mantissa (an implicity 24th bit is the leading 1 in all mantissas, which isn't stored), and the 8-bit exponent ranges from -126 to 127 with a bias of 127.
	www.cs.umass.edu /~weems/CmpSci535/535lecture6.html (3668 words)

	Number Systems
		Signed -- The representation is similar the the above unsigned binary representation, the only difference is that negative number can be represented by the first or most significant digit.
		In this from 11001010 and 01001010 are the same number 74, but 11001010 is negative or a -74 because its first digit tells the sign of the number.
		To find out what a negative number is, a number with one in its most significant bit, we invert the bits and see what the number would be positive and add a negative sign.
	home.cfl.rr.com /ravon/numbers.htm (2001 words)

	CS 2734 Computer Organization II Lecture 2 (Site not responding. Last check: 2007-10-03)
		SKILL: You should be able to write the bit patterns down for 4 bit numbers and identify their equivalent decimal representations for the four representations (unsigned, one's complement, two's complement, sign-magnitude).
		You should be able to find the representation of a number in any of the three signed number representations.
		You should be able to perform arithmetic on signed numbers in 1's complement and 2's complement.
	vip.cs.utsa.edu /classes/cs2734f97/lecture2.html (131 words)

	0 (number) Summary
		However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number.
		Zero is neither positive nor negative, neither a prime number nor a composite number, nor is it a unit.
		In some signed number representations (but not the two's complement representation predominant today) and most floating point number representations, zero has two distinct representations, one grouping it with the positive numbers and one with the negatives; this latter representation is known as negative zero.
	www.bookrags.com /0_(number) (7842 words)

	2's Complement -- Technical Notes (Site not responding. Last check: 2007-10-03)
		Negative 2's complement numbers are represented as the binary number that when added to a positive number of the same magnitude equals zero.
		To extend a signed integer from 8 bits to 16 bits or from 16 bits to 32 bits, append additional bits on the left side of the number.
		A negative number is the 7-bit binary representation of the positive number with the most significant bit set to one.
	www.evergreen.edu /biophysics/technotes/program/2s_comp.htm (440 words)

	Integral data type (Linux Reviews)
		The most common representation of a positive integer is a string of bits, using the binary numeral system.
		Twoâs complement arithmetic is convenient because there is a perfect one-to-one correspondence between representations and values, and because addition and subtraction do not need to distinguish between signed and unsigned types.
		Another, rather different, representation for integers is binary-coded decimal, which is still commonly used in mainframe financial applications and in databases.
	linuxreviews.org /dictionary/Integral_data_type (956 words)

	Number Systems
		Here is how 2-bit numbers with a sign bit work out with a sign bit on the left end of the number.
		In this representation, as in the two previous representations of negative numbers, the leftmost bit represents the sign.
		As an example, 7-bit numbers are used to represent the exponent of the power in real number representations.
	www.cs.unc.edu /~nyland/comp120/numbersystems.html (1429 words)

	DPANS94
		For all three architectures, signed numbers in the 0 to n range are bitwise identical to the corresponding unsigned number.
		Note that unsigned numbers on a signed magnitude machine are equivalent to signed non-negative numbers as a consequence of the forced correspondence between addresses and unsigned numbers and of the required behavior of + and -.
		The programmer must have a clear idea of the number of stack entries to be consumed by the execution of a word and the number of entries that will be pushed back to a stack by the execution of a word.
	www.elilabs.com /docs/forth/dpansz.html (1972 words)

	Number Systems
		Although these signed integers have solved the problem of representing positive and negative values they do not lend themselves to binary arithmetic that are performed using 'logic circuits'.
		A floating-point number is similar to scientific notation, in which a number is expressed in two parts.
		They vary in the number of bytes used, the order of the bytes, and also whether two's complement or signed integer representations are employed.
	www15.brinkster.com /dossani/numbersystem.htm (2323 words)

	Signed number representations - Wikipedia, the free encyclopedia
		One may first approach this problem of representing a number's sign by allocating one sign bit to represent the sign: set that bit (often the most significant bit) to 0 for a positive number, and set to 1 for a negative number.
		The ones' complement form of a binary number is the bitwise NOT applied to it — the complement of its positive counterpart.
		In two's complement, negative numbers are represented by the bit pattern which is one greater (in an unsigned sense) than the ones' complement of the positive value.
	en.wikipedia.org /wiki/Signed_number_representations (821 words)

	Lecture2 – Fabrication / Number Representations
		This is awkward to have this problem, and it is not easy to operate on negative numbers in this manner.
		Since this method correctly adds negative numbers, there is no reason to differentiate between addition and subtraction.
		signed answer is larger or smaller than the range of numbers represented in the word.
	www.csc.calpoly.edu /~franklin/315/review/NumberRepWorksheet.htm (437 words)

	Problem F - Signed Digit Numbers
		Computing a number from its representation in an SD system is the same as in usual number systems, just some digits have negative values.
		we can choose one of the available representations of each number in such a way that we can design an algorithm for addition which runs in constant time as the carry propagation can be eliminated.
		Given a number n in usual decimal notation fitting into a 32-bit integer, a positive b <= 10 and a satisfying the conditions stated above, you are asked to convert n into its SD(b, a) representation where the negative digits are recorded as explained above.
	acm.uva.es /p/v107/10764.html (392 words)

	Unsigned Binary
		In most cases, the number of possible values is the same as the number of possible representations.
		It's always the case that the number of values is less than or equal to the number of possible representations.
		With signed ints, you'll discover that the number of possible values is less than the number of representations.
	www.cs.umd.edu /class/sum2003/cmsc311/Notes/Data/unsigned.html (820 words)

	Function Diagrams
		One underused representation of functions, which I have found educationally rich, is the one with parallel x and y axes.
		One way to motivate many of the rules for operations with signed numbers is to generalize from patterns that work for positive numbers.
		Similar methods can be used to explore other patterns of this type, thus motivating the rules for signed number arithmetic, and providing students with a way to reconstruct them on their own.
	www.picciotto.org /math-ed/func-diag/index.html (1768 words)

	Computer Number Systems and Arithmetic (Site not responding. Last check: 2007-10-03)
		Digital computers use binary (base 2) as their `native' representation; however, in addition to binary, and decimal, it is also useful to discuss hexadecimal (base 16) and octal (base 8) representations.
		In sign-magnitude, one bit is used for sign, typically 1 = -, 0 = +; usually the most significant bit is sign, and and the low-order p-1 bits are the magnitude of the number - where we have a total of p bits.
		The floating point representation is similar to the `scientific', or `exponential' representation of very large and very small numbers on some calculators and in books; e.g.
	www.engr.udayton.edu /faculty/jloomis/ece314/notes/carch/node3.html (4363 words)

	Number Systems
		It is not efficient: the width to represent a number n is n.
		But the millinary system is still not very efficient: the radix is very large, so the number of symbols (millits) needed to represent numbers is large too.
		Signed magnitude does not lead to simple or natural arithmetic, for example, n+(-n) does not simply sum to a string of zeros.
	www.cs.fit.edu /~wds/classes/comp-org/Numbers/numbers (2355 words)

	CSCI 51A / ECE 86 Lecture Notes
		we always begin by stating n, the number of bits we are using to represent numbers, the number of significant digits
		this representation is not typically used for representation of signed integers
		the sign of the number is inherent in the representation
	www.ecst.csuchico.edu /~renner/csci221/Slides/numbers_signed (544 words)

	Introduction to Computer Organization
		The floating-point system uses a number of bytes - typically 4 or 8 - to represent the number, but with one byte (sometimes two bytes) reserved to represent the exponent e of a power-of-two multiplier for the number - the mantissa m - expressed by the remaining bytes.
		A computer's representation of integers is either perfect or only approximate, the latter situation occurring when the integer exceeds the range of numbers that a limited set of bytes can represent.
		Floating point representations have similar representation problems: if the number x can be multiplied/divided by enough powers of two to yield a fraction lying between 1/2 and 1 that has a finite binary-fraction representation, the number is represented exactly in floating point.
	cnx.org /content/m10263/latest (1829 words)

	Number representation - LearnProgramming (Site not responding. Last check: 2007-10-03)
		Quite simply, a sequence of bits--011001, for instance--is a number, but in binary form rather than the decimal form with which you’re familiar.
		Binary is nothing but an alternative numbering system to decimal, and while people use decimal all their lives, it becomes so ingrained that they can’t explain how it works and therefore can’t imagine using anything different.
		Conversion between number bases is covered, but as long as you get the idea of counting in alternative number bases, this skill is not terribly important.
	www.learnprogramming.tv /wiki/index.php?title=Number_representation (320 words)

	IEEE floating-point representations of real numbers
		A representation of the exponent is stored in the next eight bits, and the remaining twenty-three bits are occupied by a representation of the mantissa of the number.
		Such numbers are expressed in a slightly different form of scientific notation: The exponent is held fixed at -126, and the mantissa is a number greater than or equal to zero and less than one.
		Unnormalized numbers are stored less accurately than normalized ones (since there are fewer significant digits in the mantissa), but without this special convention for the all-zero exponent it would not be possible to represent them at all, and the designers of the IEEE standard felt that a degraded approximation is better than none.
	www.math.grin.edu /~stone/courses/fundamentals/IEEE-reals.html (2234 words)

	Signedness - Wikipedia, the free encyclopedia
		A numeric variable is signed if it can represent both positive and negative numbers, and unsigned if it can only represent positive numbers.
		While signed numbers can represent negative numbers they lose a range of larger numbers which can only be represented with unsigned numbers of the same size (in bits).
		This is because in signed variables, one bit is used to indicate signedness, dividing the number of positive values that can be represented by two.
	en.wikipedia.org /wiki/Signedness (174 words)

	Homework 1
		In the base 7 numbering system there are 7 digits: 0, 1, 2, 3, 4, 5, 6.
		To convert a floating point number from decimal to binary, the integer and fractional parts must be considered separately.
		The numbers to be added are given in hexadecimal: 41280000, 40700000.
	web.cs.wpi.edu /~cs2011/d06/hw1.html (702 words)

	Computers: Numbers: Representing Signed Integers
		The even number of bit patterns in a given amount of memory (eg, the 32 bits that are typically allocated for an integer) has some interesting consequences for representing signed numbers.
		If the range is symmetric (the minimum number is equal to the negative of the maximum number as in signed-magnitude and one's-complement), then there are two zeros (+0 and -0)!
		Excess notation is also used for other specialized data representations, but isn't used for general integers.
	www.leepoint.net /notes-comp/data/numbers/negatives.html (338 words)

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