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Binary numbers

Binary numbers. Primary memory. Memory = where programs and data are stored Unit = bit “BIT” is a contraction for what two words? Either a 1 or a 0 (because computers are made primarily of transistors which act as switches).

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Binary numbers

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  1. Binary numbers

  2. Primary memory • Memory = where programs and data are stored • Unit = bit • “BIT” is a contraction for what two words? • Either a 1 or a 0 (because computers are made primarily of transistors which act as switches). • Smallest addressable (i.e., can be read from or written to memory) unit is the byte. • byte = 8 bits (an 8-bit number)

  3. Binary numbers • How do computers represent numbers and do arithmetic? • How do we? What digits do we use? Why? • What is 903? • 9 is the MSD above; 3 is the LSD.

  4. Binary numbers • Can we represent numbers is other bases (radix)? • What is 11012 in base 10? • What is 110110 in base 10? • What is 11013 in base 10? • Note that they are different!

  5. Potential programming asssignment Write a program that reads in a string (not an int) representing a binary number and outputs the corresponding base 10 number. Do not use any methods from Java’s Integer class!

  6. Potential programming asssignment Write a program that reads in a string (not an int) (representing a # in base b), reads in the base b (an int), and outputs the corresponding base 10 number. (What Java String method can be used to obtain an individual char at a particular position in a string?) Do not use any methods from Java’s Integer class!

  7. Binary numbers • What are the range of digits for base 10 numbers? • For binary? • For base 5? • For some arbitrary base k? • Is 11022 a valid base 2 number? • Is 12644 a valid base 4 number?

  8. Numbers in mathematics • I = the integers • Finite or infinite? • R = the reals • Contains rational numbers like 2/7 and irrational numbers like  • Finite or infinite? • C = the complex numbers • Finite or infinite?

  9. Numbers in computers • All represented by a finite sequence of bits (1’s and 0’s). • I = the integers • Finite/bounded. • char (8 bits), short (16 bits), int (32 bits), long long (64 bits) • R = the reals • Finite/bounded. (So what about ?) • float (32 bits), double (64 bits), long double (128 bits)

  10. Numbers in computers • R = the reals • Finite/bounded. (So what about ?) • Representation is called floating point. • float (32 bits), double (64 bits), long double (128 bits) • Can also be represented by fixed point and BCD (binary coded decimal).

  11. Common ranges of values (Java) • int -2 billion to +2 billion • double 4.9x10-324 (closest to 0) to 1.8x10+308 • float 1.4x10-45 (closest to 0) to 3.4x10+38

  12. Converting from base 10 to base 2 • Convert 6110 to base 2. • Method 1: Subtract largest power of 2 repeatedly. • Also works for both ints and floats. • First, make a table of powers of 2. • 28 = 256 • 27 = 128 • 26 = 64 • 25 = 32 • 24 = 16 • 23 = 8 • 22 = 4 • 21 = 2 • 20 = 1

  13. Converting from base 10 to base 2 • Convert 6110 to base 2. • Method 1: Subtract largest power of 2 repeatedly. 1. Make a table of powers of 2. • 28 = 256 • 27 = 128 • 26 = 64 • 25 = 32 • 24 = 16 • 23 = 8 • 22 = 4 • 21 = 2 • 20 = 1 2. Repeatedly subtract (largest power of 2) msb lsb Reading forward, the result is 1111012.

  14. Use this method to convert from base 10 to base 3. • Method 1: Subtract largest powers of 3 repeatedly. Step 1: Create table of powers of 3. • 34 = 81 • 33 = 27 • 32 = 9 • 31 = 3 • 30 = 1 Step 2: Repeatedly subtract (largest) multiples of largest power of 3. (Why? Recall digit range for a particular base.)

  15. Use this method to convert from base 10 to base 3. • Method 1: Subtract largest powers of 3 repeatedly. Step 1: Create table of powers of 3. • 34 = 81 • 33 = 27 • 32 = 9 • 31 = 3 • 30 = 1 Step 2: Repeatedly subtract (largest) multiples of largest power of 3. • Convert 6110 to base 3. msd lsd Reading forward, the result is 20213.

  16. Potential programming assignment Write a program that takes as input a base 10 number (an int) and outputs the corresponding base 2 number. Do I need to remind you again? Do not use any methods from Java’s Integer class!

  17. Converting base 10 floating point numbers to base 2 using Method 1 (repeated subtraction) Step 1: Create table of powers of 2. 24 = 16 23 = 8 22 = 4 21 = 2 20 = 1 2-1 = 0.5 2-2 = 0.25 2-3 = 0.125 2-4 = 0.0625 2-5 = 0.03125 2-6 = 0.015625 Step 2: Subtract largest power of 2 repeatedly.

  18. Converting base 10 floating point numbers to base 2 using Method 1 (repeated subtraction) Step 1: Create table of powers of 2. 24 = 16 23 = 8 22 = 4 21 = 2 20 = 1 2-1 = 0.5 2-2 = 0.25 2-3 = 0.125 2-4 = 0.0625 2-5 = 0.03125 2-6 = 0.015625 Step 2: Subtract largest power of 2 repeatedly. Convert 11.07812510 to base 2. Reading forward, the result is 1011.0001012.

  19. Conversion methods Method 1. Using a table, repeatedly subtract largest multiple. - directly works for both ints and floats - requires a table Method 2: Repeated (integer) division (with remainder) by desired base. - works for ints but needs to be “modified” to work for floats - doesn’t require a table

  20. Method 2: Repeated (integer) division (with remainder) by desired base. Convert 6110 to base 2. lsb msb Reading backwards, the result is 1111012.

  21. Method 2: Repeated (integer) division (with remainder) by desired base. Convert 6110 to base 3. lsd msd Reading backwards, the result is 20213.

  22. Method 2 for fp numbers • Convert 61.687510 to base 2. • We just saw how to convert 6110 to base 2 (by repeated division with desired base). • So all we need to do is convert 0.687510 to base 2. • This can (analogously) be accomplished by repeated multiplication by the desired base.

  23. Method 2 • Convert 0.687510 to base 2 by repeated multiplication by desired base. • Then read integer part forwards.

  24. Method 2 • Convert 0.110 to base 2 by repeated multiplication by desired base. • Then read integer part forwards. Do you see the repetition/loop?

  25. Method 2 • Convert 0.110 to base 2 by repeated multiplication by desired base. • Then read integer part forwards. Do you see the repetition/loop? same same

  26. Converting between arbitrary bases • In general, to convert from base u to base v, we typically: • Convert from base u to base 10. • Then convert from base 10 to base v. • When converting from base 2 to two other important bases, we can skip the intermediate conversion through base 10 and convert directly from base 2 to these two other bases (and vice versa).

  27. Other important bases related to base 2: base 8 • Octal, base 8 • Digits: 0..7 base 8 base 2 0 000 1 001 2 010 3 011 4 100 5 101 6 110 7 111 • Problems: • Convert 1008 to base 10. • Convert 1008 to base 2. • Convert 11012 to base 10. • Convert 11012 to base 8. • Convert 7028 to base 10. • Convert 7028 to base 2. • Convert 8028 to base 2.

  28. Other important bases related to base 2: base 16 • Hex (hexadecimal), base 16 • Digits: 16 of them (more than base 10) so we’ll will need 6 extra symbols: {0..9,a..f} base 16 base 10 base 2 base 8 0 0 0000 00 1 1 0001 01 2 2 0010 02 3 3 0011 03 4 4 0100 04 5 5 0101 05 6 6 0110 06 7 7 0111 07 8 8 1000 10 9 9 1001 11 a 10 1010 12 b 11 1011 13 c 12 1100 14 d 13 1101 15 e 14 1110 16 f 15 1111 17

  29. Problems: • Convert 7258 to base 2 to base 16 to base 10.

  30. Problems: • Convert 7258 to base 2 to base 16 to base 10. • 111 010 1012 • 1 1101 01012 = 1d516 • 1x162 + 13x161 + 5x160 = 46910

  31. Convert 153.51310 to base 8.Approach: convert 153 and 0.513 separately. remainder So the result is 231.40651… base 8.

  32. Problem: Convert 3115 to X10 to Y2. • 3x52+1x51x1x50 = 75+5+1 = 8110.

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