Binary Numbers Material on Data Representation can be found in Chapter 2 of Computer Architecture (Nicholas Carter) CSIT 301 (Blum)

Why Binary? Maximal distinction among values  minimal corruption from noise. Imagine taking the same physical attribute of a circuit, e.g. a voltage lying between 0 and 5 volts, to represent a number. The overall range can be divided into any number of regions. CSIT 301 (Blum)

Don’t sweat the small stuff For decimal numbers, fluctuations must be less than 0.25 volts. For binary numbers, fluctuations must be less than 1.25 volts. 5 volts 0 volts Decimal Binary CSIT 301 (Blum)

Range actually split in three High Forbidden range Low CSIT 301 (Blum)

It doesn’t matter …. Two of the standard voltages coming from a computer’s power supply are ideally supposed to be 5.00 volts and 12.00 volts Measurements often reveal values that are slightly off – e.g. 5.14 volts or 12.22 volts or some such value. So what, who cares. CSIT 301 (Blum)

How to represent big integers Use positional weighting, same as with decimal numbers 205 = 2102 + 0101 + 5100 11001101 = 127 + 126 + 025 + 024 + 123 + 122 + 021 + 120 = 128 + 64 + 8 + 4 + 1 = 205 CSIT 301 (Blum)

Converting 205 to Binary 205/2 = 102 with a remainder of 1, place the 1 in the least significant digit position Repeat 102/2 = 51, remainder 0 1 1 CSIT 301 (Blum)

Iterate 51/2 = 25, remainder 1 25/2 = 12, remainder 1 1 1 CSIT 301 (Blum)

Iterate 6/2 = 3, remainder 0 3/2 = 1, remainder 1 1/2 = 0, remainder 1 1 1 1 CSIT 301 (Blum)

Recap 205 1 127 + 126 + 025 + 024 + 123 + 122 + 021 + 120 CSIT 301 (Blum)

Finite representation Typically we just think computers do binary math. But an important distinction between binary math in the abstract and what computers do is that computers are finite. There are only so many flip-flops or logic gates in the computer. When we declare a variable, we set aside a certain number of flip-flops (bits of memory) to hold the value of the variable. And this limits the values the variable can have. CSIT 301 (Blum)

Same number, different representation 5 using 8 bits 0000 0101 5 using 16 bits 0000 0000 0000 0101 5 using 32 bits 0000 0000 0000 0000 0000 0000 0000 0101 CSIT 301 (Blum)

Adding Binary Numbers Same as decimal; if the sum of digits in a given position exceeds the base (10 for decimal, 2 for binary) then there is a carry into the next higher position 1 3 9 + 5 7 4 CSIT 301 (Blum)

Adding Binary Numbers 1 + carries 39 35 74 CSIT 301 (Blum)

Uh oh, overflow What if you use a byte (8 bits) to represent an integer A byte may not be enough to represent the sum of two such numbers. 1 170 204 118??? CSIT 301 (Blum)

Biggest unsigned* integers 4 bit: 1111  15 = 24 - 1 8 bit: 11111111  255 = 28 – 1 16 bit: 1111111111111111  65535= 216 – 1 32 bit: 11111111111111111111111111111111  4294967295= 232 – 1 Etc. *If one uses all of the bits available to represent only positive counting numbers, one is said to be working with unsigned integers. CSIT 301 (Blum)

Bigger Numbers High-level languages often offer a hierarchy of types that differ in the number of bits used. You can represent larger numbers than allowed by the highest type in the hierarchy by using more words. You just have to keep track of the overflows to know how the lower numbers (less significant words) are affecting the larger numbers (more significant words). CSIT 301 (Blum)

Negative numbers Negative x is the number that when added to x gives zero Ignoring overflow the two eight-bit numbers above add up to zero 1  x  -x CSIT 301 (Blum)

Two’s Complement: a two-step procedure for finding -x from x Step 1: exchange 1’s and 0’s Step 2: add 1 (to the lowest bit only) 1  x 1 1  -x CSIT 301 (Blum)

Sign bit With the two’s complement approach, all positive numbers start with a 0 in the left-most, most-significant bit and all negative numbers start with 1. So the first bit is called the sign bit. But note you have to work harder than just strip away the first bit. 10000001 IS NOT the 8-bit version of –1 CSIT 301 (Blum)

Add 1’s to the left to get the same negative number using more bits -5 using 8 bits 11111011 -5 using 16 bits 1111111111111011 -5 using 32 bits 11111111111111111111111111111011 When the numbers represented are whole numbers (positive or negative), they are called just integers or signed. CSIT 301 (Blum)

3-bit signed and unsigned 7 1 6 5 4 3 2 3 1 2 -1 -2 -3 -4 Think of driving a brand new car in reverse. What would happen to the odometer? CSIT 301 (Blum)

Biggest signed integers 4 bit: 0111  7 = 23 - 1 8 bit: 01111111  127 = 27 – 1 16 bit: 0111111111111111  32767= 215 – 1 32 bit: 01111111111111111111111111111111  2147483647= 231 – 1 Etc. CSIT 301 (Blum)

Most negative signed integers 4 bit: 1000  -8 = - 23 8 bit: 10000000  - 128 = - 27 16 bit: 1000000000000000  -32768= - 215 32 bit: 10000000000000000000000000000000  -2147483648= - 231 Etc. CSIT 301 (Blum)

Riddle 1 Is it 214? Or is it – 42? Or is it Ö? Or is it …? It’s a matter of interpretation How was it declared? 1 CSIT 301 (Blum)

Hexadecimal Numbers Even moderately sized decimal numbers end up as long strings in binary. Hexadecimal numbers (base 16) are often used because the strings are shorter and the conversion to binary is easier. There are 16 digits: 0-9 and A-F. CSIT 301 (Blum)

Decimal  Binary  Hex 0  0000  0 1  0001  1 2  0010  2 3  0011  3 4  0100  4 5  0101  5 6  0110  6 7  0111  7 8  1000  8 9  1001  9 10  1010  A 11  1011  B 12  1100  C 13  1101  D 14  1110  E 15  1111  F CSIT 301 (Blum)

Binary to Hex Break a binary string into groups of four bits (nibbles). Convert each nibble separately. 1 E C 9 CSIT 301 (Blum)

Numbers from Logic All of the numerical operations we have talked about are really just combinations of logical operations. E.g. the adding operation is just a particular combination of logic operations Possibilities for adding two bits 0+0=0 (with no carry) 0+1=1 (with no carry) 1+0=1 (with no carry) 1+1=0 (with a carry) CSIT 301 (Blum)

Addition Truth Table INPUT OUTPUT A B Sum A XOR B Carry A AND B 1 1 CSIT 301 (Blum)

Multiplication: Shift and add 1  + shift shift CSIT 301 (Blum)

Fractions Similar to what we’re used to with decimal numbers 3.14159 = 3 · 100 + 1 · 10-1 + 4 · 10-2 + 1 · 10-3 + 5 · 10-4 + 9 · 10-5 11.001001 = 1 · 21 + 1 · 20 + 0 · 2-1 + 0 · 2-2 + 1 · 2-3 + 0 · 2-4 + 0 · 2-5 + 1 · 2-6 (11.001001  3.140625) CSIT 301 (Blum)

Converting decimal to binary II 98.61 Integer part 98 / 2 = 49 remainder 0 49 / 2 = 24 remainder 1 24 / 2 = 12 remainder 0 12 / 2 = 6 remainder 0 6 / 2 = 3 remainder 0 3 / 2 = 1 remainder 1 1 / 2 = 0 remainder 1 1100010 CSIT 301 (Blum)

Converting decimal to binary III 98.61 Fractional part 0.61  2 = 1.22 0.22  2 = 0.44 0.44  2 = 0.88 0.88  2 = 1.76 0.76  2 = 1.52 0.52  2 = 1.04 .100111 CSIT 301 (Blum)

Another Example (Whole number part) 123.456 Integer part 123 / 2 = 61 remainder 1 61 / 2 = 30 remainder 1 30 / 2 = 15 remainder 0 15 / 2 = 7 remainder 1 7 / 2 = 3 remainder 1 3 / 2 = 1 remainder 1 1 / 2 = 0 remainder 1 1111011 CSIT 301 (Blum)

Checking: Go to All Programs/Accessories/Calculator PHY 201 (Blum)

Put the calculator in Programmer view PHY 201 (Blum)

Enter number, put into binary mode PHY 201 (Blum)

Another Example (fractional part) 123.456 Fractional part 0.456  2 = 0.912 0.912  2 = 1.824 0.824  2 = 1.648 0.648  2 = 1.296 0.296  2 = 0.592 0.592  2 = 1.184 0.184  2 = 0.368 … .0111010… CSIT 301 (Blum)

Convert to decimal mode, then PHY 201 (Blum)

Edit/Copy result. Switch to Scientific View. Edit/Paste PHY 201 (Blum)

Divide by 2 raised to the number of digits (in this case 7, including leading zero) 1 2 3 4 PHY 201 (Blum)

Finally hit the equal sign. In most cases it will not be exact PHY 201 (Blum)

Other way around Multiply fraction by 2 raised to the desired number of digits in the fractional part. For example .456  27 = 58.368 Throw away the fractional part and represent the whole number 58 111010 But note that we specified 7 digits and the result above uses only 6. Therefore we need to put in the leading 0 0111010 CSIT 301 (Blum)

Fixed point If one has a set number of bits reserved for representing the whole number part and another set number of bits reserved for representing the fractional part of a number, then one is said to be using fixed point representation. The point dividing whole number from fraction has an unchanging (fixed) place in the number. CSIT 301 (Blum)

Limits of the fixed point approach Suppose you use 4 bits for the whole number part and 4 bits for the fractional part (ignoring sign for now). The largest number would be 1111.1111 = 15.9375 The smallest, non-zero number would be 0000.0001 = .0625 CSIT 301 (Blum)

Floating point representation Floating point representation allows one to represent a wider range of numbers using the same number of bits. It is like scientific notation. CSIT 301 (Blum)

Scientific notation Used to represent very large and very small numbers. Ex. Avogadro’s number  6.0221367  1023 particles  602213670000000000000000 Ex. Fundamental charge e  1.60217733  10-19 C  0.000000000000000000160217733 C CSIT 301 (Blum)

Scientific notation: all of these are the same number 12345.6789 = 1234.56789  100 1234.56789  10 = 1234.56789  101 123.456789  100 =123.456789  102 12.3456789  103 1.23456789  104 Rule: Shift the point to the left and increment the power of ten. CSIT 301 (Blum)

Small numbers 0.000001234 0.00001234  10-1 0.0001234  10-2 0.001234  10-3 0.01234  10-4 0.1234  10-5 1.234  10-6 Rule: shift point to the right and decrement the power. CSIT 301 (Blum)

Floating Point Rules We’ll use a set of rules that are close but not quite the same as the IEEE 754 standards for floating point representation. Starting with the fixed point binary representation, shift the point and increase the power (of 2 now that we’re in binary). Shift so that the number has no whole number part and also so that the first fractional bit (the half’s place) has a 1. CSIT 301 (Blum)

Floats SHIFT expression so it is just under 1 and keep track of the number of shifts 1100010.1001100110011001 .11000101001100110011001  27 Express the number of shifts in binary .11000101001100110011001  200000111 We’re not done yet so this exponent will change. CSIT 301 (Blum)

Mantissa and Exponent and Sign .11000101001100110011001  200000111 (Significand) Mantissa Exponent The number may be negative, so there a bit (the sign bit) reserved to indicate whether the number is positive or negative CSIT 301 (Blum)

Small numbers 0.000010101110 0.10101110  2-4 The power (a.k.a. the exponent) could be negative so we have to be able to deal with that. Floating point numbers use a procedure known as biasing to handle the negative exponent problem. CSIT 301 (Blum)

Biasing Actually the exponent is not represented as shown on the previously. There were 8 bits used to represent the exponent on the previous slide, that means there are 256 numbers that could be represented. Since the exponent could be negative (to represent numbers less than 1), we choose half of the range to be positive and half to be negative , i.e. -128 to 127. CSIT 301 (Blum)

Biasing (Cont.) In biasing, one does not use 2’s complement or a sign bit. Instead one adds a bias (equal to the magnitude of the most negative number) to the exponents and represents the result of that addition. CSIT 301 (Blum)

Biasing (Cont.) With 8 bits, the bias is 128 (= 27 that is 2 raised to the number of bits used for the exponent minus one). In our previous example, we had to shift 7 times to the left, corresponding to an exponent of +7. We add that shift to the bias 128+7=135. That is the number we put in the exponent portion: 135  10000111. CSIT 301 (Blum)

Big floats Assume we use 8 bits, 4 for the mantissa and 4 for the exponent (neglecting sign). What is the largest float? Mantissa: 1111 Exponent 1111 0.9375  27 =120 (Compare this to the largest fixed-point number using the same amount of space 15.9375) CSIT 301 (Blum)

Small floats Assume we use 8 bits, 4 for the mantissa and 4 for the exponent (neglecting sign). What is the smallest float? Mantissa: 1000 Exponent 0000 0.5  2-8 = 0.001953125 (Compare this to the smallest fixed-point number using the same amount of space .0625) CSIT 301 (Blum)

Adding Floats Consider adding the following numbers expressed in scientific notation 3.456789  103 1.212121  10-2 The first step is to re-express the number with the smaller magnitude so that it has the same exponent as the other number. CSIT 301 (Blum)

Adding Floats (Cont.) 1.212121  10-2 0.1212121  10-1 0.01212121  100 0.001212121  101 0.0001212121  102 0.00001212121  103 The number was shifted 5 times (3-(-2)). CSIT 301 (Blum)

Adding Floats (Cont.) When the exponents are equal the mantissas can be added. 3.456789  103 0.00001212121  103 =3.45680112121  103 CSIT 301 (Blum)

Rounding In a computer there are a finite number of bits used to represent a number. When the smaller floating-point number is shifted to make the exponents equal, some of the less significant bits are lost. This loss of information (precision) is known as rounding. CSIT 301 (Blum)

One more fine point about floating-point representation As discussed so far, the mantissa (significand) always starts with a 1. When storage was expensive, designers opted not to represent this bit, since it is always 1. It had to be inserted for various operations on the number (adding, multiplying, etc.), but it did not have to be stored. CSIT 301 (Blum)

Still another fine point When we assume that the mantissa must start with a 1, we lose 0. Zero is too important a number to lose, so we interpret the mantissa of all zeros and exponent of all zeros as zero Even though ordinarily we would assume the mantissa started with a one that we didn’t store. CSIT 301 (Blum)

Yet another fine point In the IEEE 754 format for floats, you bias by one less (127) and reserve the exponents 00000000 and 11111111 for special purposes. One of these special purposes is “Not a number” (NaN) which is the floating point version of overflow. CSIT 301 (Blum)

An example Represent -9087.8735 as a float using 23 bits for the mantissa, 8 for the exponent and one for the sign. Convert the whole number magnitude 9087 to binary: 10 0011 0111 1111 That uses up 14 of the 23 bits for the mantissa, leaving 9 for the fractional part. CSIT 301 (Blum)

An example (Cont.) Multiply the fractional part by 29 and convert whole number part of that to binary, make sure in uses 9 bits (add leading 0’s if it doesn’t). .8735  29 = 447.232 447  110111111 CSIT 301 (Blum)

An example (Cont.) 10001101111111.110111111 .10001101111111110111111  214 Mantissa 10001101111111110111111 Exponent 14+128=142  10001110 Sign bit 1 (because number was negative) CSIT 301 (Blum)

Example 2 0.0076534 No whole number part. Begin by using all 23 mantissa bits for the fractional part. 0.0076534  223 = 64201.3724672 64201  1111101011001001 Only uses 16 places, means that so far number starts with 7 zeros. But float mantissas are supposed to start with 1. CSIT 301 (Blum)

Example 2 (Cont.) 23+7 0.0076534  230 = 8217775.6758016 821775  11111010110010010101111 Above is mantissa Exponent 128 – 7 = 121  01111001 Sign bit 0 (positive number) CSIT 301 (Blum)

References Computer Architecture, Nicholas Carter Computer Systems: Organization and Architecture, John Carpinelli CSIT 301 (Blum)