If the numbers being added are of opposite signs then an overflow will never

# If the numbers being added are of opposite signs then

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If the numbers being added are of opposite signs, then an overflow will never occur. As an alternative method of detecting overflow for addition, an overflow occurs if and only if the carry into the sign bit differs from the carry out of the sign bit. 22
Cont… Consider adding (+80)10 and (+50)10 using an eight bit format. The result should be (+130)10, however, as shown below, the result is (- 126)10: + 0 1 0 1 0 0 0 0 (+80)10 + 0 0 1 1 0 0 1 0 (+50)10 ——————— + 1 0 0 0 0 0 1 0 (-126)10 This should come as no surprise, since we know that the largest positive 8-bit two’s complement number is +(127)10, and it is therefore impossible to represent (+130)10. Although the result (10000010)2 “looks” like (130)10 if we think of it in unsigned form, the sign bit indicates a negative number in the signed form, which is clearly wrong. 23
24 Exercise Determine the decimal value represented by 10001011 in each of the following four systems. 1. Unsigned notation? 2. Signed magnitude notation? 3. One’s complement 4. Tow’s complements?
25 2.4.2 Fraction Representation To represent fraction we need other representations: Fixed point representation Floating point representation.
26 2.4.2.1 Fixed-Point Representation old position 7 6 5 4 3 2 1 0 New position 4 3 2 1 0 -1 -2 -3 Bit pattern 1 0 0 1 1 . 1 0 1 Contribution 2 4 2 1 2 0 2 -1 2 -3 =19.625 Radix-point
27 Limitation of Fixed-Point Representation To represent large numbers or very small numbers we need a very long sequences of bits. This is because we have to give bits to both the integer part and the fraction part.
28 Floating Point -15900000000000000 could be represented as - 15.9 * 10 15 - 1.59 * 10 16 A calculator might display 159 E14 - 159 * 10 14 Sign Exponent Base Mantissa
29 M B E Sign mantissa or base exponent significand Floating point format Sign Exponent Mantissa
30 Example 32 bit floating representation The number will occupy 32 bits The first bit represents the sign of the number; 1= negative 0= positive. The next 8 bits will specify the exponent The remaining 23 bits will carry the mantissa normalised to be between
Normalization A potential problem with representing floating point numbers is that the same number can be represented in different ways, which makes comparisons and arithmetic operations difficult. For example, consider the numerically equivalent forms shown below: 3584.1 × 100 = 3.5841 × 103 = .35841 × 104. In order to avoid multiple representations for the same number, floating point numbers are maintained in normalized form. That is, the radix point is shifted to the left or to the right and the exponent is adjusted accordingly until the radix point is to the left of the leftmost nonzero digit. So the rightmost number above is the normalized one. Unfortunately, the number zero cannot be represented in this scheme, so to represent zero an exception is made. The exception to this rule is that zero is represented as all 0’s in the mantissa 31
If the mantissa is represented as a binary, that is, base 2, number, and if the normalization condition is that there is a leading “1” in the normalized mantissa, then there is no need to store that “1” and in fact, most floating point formats do not store it. Rather, it is “chopped off ” before packing up the number for storage, and it is restored when unpacking the number into exponent and mantissa.

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• Fall '19
• Binary numeral system, Decimal, mantissa
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