Integer Representation
- 1. Software Developers View of Hardware Integer Representation
- 2. Syllabus Integer Representation, including: Sign and modulus One’s complement Two’s complement Representation of Fractions Floating point or real. IEEE
- 3. What is Integer Representation About? To fully understand how binary numbers are processed we must have an understanding of how integers are handled by the computer. For example: How are negative numbers represented? How are floating point numbers represented? Processing – How do we add, subtract, divide and multiply?
- 4. How can we represent a binary number? ?????? Maybe - 000111001 NO
- 5. How can we represent a binary number? Binary number can only be represented by a 0 or 1, so we cannot use a (-) to denote a negative number. Therefore, we cannot use a specialised symbol so it must be either a 0 or 1. There are THREE methods used, they are:
- 6. Sign and Modulus This method uses the far left-hand bit which is often referred to as the sign bit. The general rule is that a positive number starts with 0 and a negative number starts with a 1. + 28 = 0 0011100 SIGN Modulus
- 7. ACTIVITY 1 Using ONE byte what is the maximum positive and negative integer that can represented using sign and modulus. Convert the following numbers to binary using sign and modulus. 122 -100 -264
- 8. One’s Complement What is a complement ? It is the opposite of something. Because computers do not like to subtract, this method finds the complement of a positive number and then addition can take place. -127 -75 0 75 127
- 9. One’s Complement Example Find the one’s complement of +100 Convert +100 to binary. Swap all the bits. Check answer by adding -127 to your result.
- 10. One’s Complement Example +100 = 01100100 2 01100100 2 10011011 2 10011011 2 = -27 10 127 10 + -27 10 = 100 10 127 -127 65 -62 -27 100
- 11. ACTIVITY 2 Convert the following binary numbers into a negative using one’s complement. 0110111 2 00110011 2
- 12. Two’s Complement The only difference between the two processes is that the left most bit is -128 10 rather than -127 10 . The process for two’s complement is exactly the same as one’s complement, however you are then required to add 1 to the results.
- 13. Two’s Complement Example Find the two’s complement of +15 10 Convert +15 10 to binary. Swap all the bits. Add 1 to the result.
- 14. Two’s Complement Example +15 = 00001111 2 00001111 2 11110000 2 11110000 2 + 1 11110001 -128 + 64 + 32 + 16 + 1 = -15 10
- 15. Two’s Complement Rules for Adding 1 + 1 = 0 (Carry the 1) 0 + 0 = 0 0 + 1 = 1
- 16. ACTIVITY 3 Convert the following binary numbers into a negative using two’s complement. 00101110 2 01101000 2 01011011 2 A computer has a word length of 8 binary digits and uses two’s complement. Translate -82 10 into a form a computer would use.
- 17. Answers 00101110 - 11010010 01101000 - 10011000 01011011 - 10100101
- 18. ANSWER Translate 82 10 into binary = 01010010 2 Reverse all the bits so that 0’s become 1’s and 1’s become 0’s. 10101101 = -82 10 in one’s complement. Complete two’s complement. Answer = 10101110 2
- 19. Representation of Fractions What are they? Fractional numbers are numbers between 0 and 1. They are called real numbers in computing terms. Real numbers contain both an integer and fractional part. E.g. 23.714 OR 01101.1001
- 20. Methods Used: Computers use two main methods to represent real numbers: They are: Fixed point Floating point
- 21. Fixed Point This method assumes the decimal point is in a fixed position. Relies on using a fixed number of bits for both the integer and fractional parts i.e. 5 and 3. Example 10011101 = 10011.101
- 22. Fixed Point Therefore 16 + 2 + 1 = 19 (Integer) 0.5 + 0.125 = 0.625 (Fraction) So 10011.101 = 19.625 1 0 1 1 1 0 0 1 0.125 0.25 0.5 1 2 4 8 16 Fractional Integer
- 23. Fixed Point Problems are: You fix the numbers that you can represent i.e. you are limited to amount of numbers that you can actually represent.
- 24. Activity 4 Convert the following binary numbers to decimal. 101.11 110.101 Convert the following decimal numbers to binary. 2.25 0.875
- 25. Activity 4 Identify what number 11001101 would represent in the integer /fractional ratio was: a. 6:2 b. 5:3 c: 4:4
- 26. Floating Point This is the preferred method because you can represent large numbers. This uses exponential notation which highlights two specific parts of a decimal number: Mantissa – Fractional part. Exponent – Is the power by 10 which the mantissa is multiplied.
- 27. Floating Point Example -241.65 = -0.24165 x 10 3 0.0028 = 0.28 x 10 -2 110.11 = 0.11011 x 2 3 The aim of floating point representation is show how many numbers before or after the decimal point.
- 28. Activity 5 Identify the floating point representation of the following numbers. 3776.56 10001.11 0.0000010100 10.001001
- 29. Floating Point But there is more! A computer will represent a binary number into THREE parts: Sign Bit Mantissa Exponent
- 30. Floating Point Convert the following binary number to decimal using floating point notation. 01101011 Sign Bit Mantissa Exponent
- 31. Floating Point Steps Convert the mantissa to decimal. 0.5 + 0.25 + 0.0625 = 0.8125 1 0.0625 0 1 1 0.125 0.25 0.5
- 32. Floating Point Steps Convert the exponent. 011 = 3 Therefore 2 3 = 8 0.8125 x 8 = 6.5 Answer = 6.5
- 33. Floating Point More on Floating Point Numbers The Institute of Electrical and Electronics Engineers has produced a standard for representing binary numbers using the floating point system. IEEE 32 bit IEEE 64 bit
- 34. IEEE 32 Bit 1 11001010 00000000000000000011001 32 Bits Sign Bit (1 bit) Exponent (8 bits) Mantissa (23 bits)
- 35. IEEE 32 Bit Example – Convert 24.25 to an IEEE 32 bit number. Convert 24.25 to binary. 24.25 = 11000.01 Normalise 11000.01 = 1.100001 x 2 4 Determine the sign bit, mantissa and exponent.
- 36. IEEE 32 Bit Sign Bit = 0 (Because it is a + number) Mantissa = 100001 Exponent = 127 (Largest) + 4 = 131 0 10000011 00000000000000000010001
- 37. Activity 6 Convert the following number using IEEE 32 bit format. 418.125