This document provides an overview of different number systems including decimal, binary, octal, and hexadecimal. It explains how each system uses different bases and sets of digits. The key concepts covered include:
- How each number system represents values using place-value and place-weighting
- Techniques for converting between decimal, binary, octal, and hexadecimal numbers using either successive division or weighted multiplication
- How fractional numbers are represented in different bases using a radix point
- The significance of place-value for the most and least significant digits
This document provides an overview of different number systems including decimal, binary, octal, and hexadecimal. It explains how each system uses different bases and sets of digits. The key concepts covered include:
- How each number system represents values using place-value and place-weighting
- Techniques for converting between decimal, binary, octal, and hexadecimal numbers using either successive division or weighted multiplication
- How fractional numbers are represented in different bases using a radix point
- The significance of place-value for the most and least significant digits
This document provides an overview of different number systems including decimal, binary, octal, and hexadecimal. It explains how each system uses different bases and sets of digits. The key concepts covered include:
- How each number system represents values using place-value and place-weighting
- Techniques for converting between decimal, binary, octal, and hexadecimal numbers using either successive division or weighted multiplication
- How fractional numbers are represented in different bases using a radix point
- The significance of place-value for the most and least significant digits
This document provides an overview of different number systems including decimal, binary, octal, and hexadecimal. It explains how each system uses different bases and sets of digits. The key concepts covered include:
- How each number system represents values using place-value and place-weighting
- Techniques for converting between decimal, binary, octal, and hexadecimal numbers using either successive division or weighted multiplication
- How fractional numbers are represented in different bases using a radix point
- The significance of place-value for the most and least significant digits
Dr. A. Sahu Dept of Comp. Sc. & Engg. Dept of Comp. Sc. & Engg. Indian Institute of Technology Guwahati
1 Outline • Number System – Decimal, Binary, Octal, Hex D i l Bi O t l H • Conversion (one to another) – Decimal to Binary, Octal, Hex & Vice Versa Decimal to Binary Octal Hex & Vice Versa – Binary to HEX & vice versa • Other representation – Signed, Unsigned, Complement • Operation – Add, Sub, Mul, Div, Mod • How to handle real number efficiently ? – Float, Double Float Double 2 What Digit? =>> Number System What Digit? Number System • Famous Number System: Dec, Rom, Bin Famous Number System: Dec Rom Bin • Decimal System: 0 ‐9 – May evolves: because human have 10 finger M l b h h 10 fi • Roman System – May evolves to make easy to look and feel – Pre/Post Concept: (IV, V & VI) is (5‐1, 5 & 5+1) • Binary System, Others (Oct, Hex) – One can cut an apple in to two pp
Most significant digit Least significant digit Least significant digit
Decimal: Decimal: 1063079
Most significant digit
Most significant digit Least significant digit Least significant digit Decimal (base 10) Decimal (base 10) • Uses Uses positional representation positional representation • Each digit corresponds to a power of 10 based on its position in the number on its position in the number • The powers of 10 increment from 0, 1, 2, etc. as you move right to left i h l f – 1,479 = 1 * 103 + 4 * 102 + 7 * 101 + 9 * 100 Binary (base 2) Binary (base 2)
• Two digits: 0, 1 • To make the binary numbers more y readable, the digits are often put in groups of 4 – 1010 = 1 * 23 + 0 * 22 + 1 * 21 + 0 * 20 = 8 + 2 = 10 = 10 – 1100 1001 = 1 * 27 + 1 * 26 + 1 * 23 + 1 * 20 = 128 128 + 64 64 + 8 8 + 1 1 = 201 How to Encode Numbers: Binary Numbers • Working with binary numbers – In base ten, helps to know powers of 10 In base ten helps to know powers of 10 • one, ten, hundred, thousand, ten thousand, ... – In base two, helps to know powers of 2 I b t h l t k f2 • one, two, four, eight, sixteen, thirty two, sixty f four, one hundred twenty eight h d dt t i ht • Count up by powers of two
Counting 0 1 00000 00001 0 1 0 1 2 00010 2 2 3 00011 3 3 4 00100 4 4 5 00101 5 5 6 00110 6 6 7 00111 7 7 8 01000 10 8 9 01001 11 9 10 01010 12 A 11 01011 13 B 12 01100 14 C 13 01101 15 D 14 01110 16 E 15 01111 17 F 10 16 10000 20 10 Fractional Number Fractional Number • Point: Decimal Point, Binary Point, Hexadecimal point • Decimal 247.75 = 2x102+4x101+7x100+7x10‐1+5x10‐2 • Binary 10.101= 1x21+0x20+1x2‐1+0x2‐2+1x2‐3 • Hexadecimal 6A.7D=6x161+10x160+7x16‐1+Dx16‐2 Converting To and From Decimal Converting To and From Decimal
Successive Decimal10 0 1 2 3 4 5 6 7 8 9 Weighted Division Weighted Successive Multiplication M lti li ti Multiplication Division Successive Weighted Di i i Division M lti li ti Multiplication Octal8 Hexadecimal16 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 A B C D E F Binary2 0 1
a)) Divide the decimal number by 2; the remainder is the LSB of the binary y ; y number. b) If the quotation is zero, the conversion is complete. Otherwise repeat step (a) using the quotation as the decimal number. The new remainder is the ( ) g q next most significant bit of the binary number.
Weighted Multiplication
a) Multiply each bit of the binary number by its corresponding bit‐ weighting factor (i.e., Bit‐0→20=1; Bit‐1→21=2; Bit‐2→22=4; etc). b) Sum up all of the products in step (a) to get the decimal number. Decimal to Binary : Subtraction Method y • Goal Desired decimal number: 12 Desired decimal number: 12 – Good for human 32 16 8 4 2 1 – Get the binary weights to =32 1 add up to the decimal dd t th d i l 32 16 8 4 2 1 too much quantity 0 1 =16 • Work from left to right 32 16 8 4 2 1 too much a • (Right to left – may fill in 1s =8 0 0 1 that shouldn’t have been 32 16 8 4 2 1 ok, keep going there – try it). 0 0 1 1 =8+4=12 32 16 8 4 2 1 DONE
0 0 1 1 0 0 answer 32 16 8 4 2 1 Decimal to Binary : Division Method y • Good for computer: Divide decimal number by 2 and insert remainder into new binary number. y – Continue dividing quotient by 2 until the quotient is 0. • Example: Convert decimal number 12 to binary
12 div 2 = ( Quo=6 , Rem=0) LSB 6 div 2 = (Quo=3, Rem=0) ( ) 3 div 2 = (Quo=1,Rem=1) 1 div 2 = ( Quo=0 Rem=1) MSB 1 div 2 = ( Quo=0, Rem=1) MSB
a)) Divide the decimal number by N; the remainder is the LSB of the ANY y ; BASE Number . b) If the quotient is zero, the conversion is complete. Otherwise repeat step (a) using the quotient as the decimal number. The new remainder p( ) g q is the next most significant bit of the ANY BASE number.
Weighted Multiplication
a) Multiply each bit of the ANY BASE number by its corresponding bit‐ weighting factor (i.e., Bit‐0→N0; Bit‐1→N1; Bit‐2→N2; etc). b) Sum up all of the products in step (a) to get the decimal number. Decimal ↔ Octal Conversion Decimal ↔ Octal Conversion The Process: Successive Division • Divide number by 8; R is the LSB of the octal number • While Q is 0 • Using the Q as the decimal number. • New remainder is MSB of the octal number. 11 8 94 r = 6 ← LSB
1 9410 = 1368 8 11 r =3
0 8 1 r = 1 ← MSB 17 Decimal ↔ Hexadecimal Conversion Decimal ↔ Hexadecimal Conversion The Process: Successive Division • Divide number by 16; R is the LSB of the hex number • While Q is 0 • Using the Q as the decimal number. • New remainder is MSB of the hex number. 5 16 94 r = E ← LSB S
0 9410 = 5E16 16 5 r = 5 ← MSB Example: Hex → Octal Example: Hex → Octal Example: C Convert the hexadecimal number 5A h h d i l b 5AH into its octal equivalent. i i l i l Solution: First convert the hexadecimal number into its decimal equivalent, then First convert the hexadecimal number into its decimal equivalent then convert the decimal number into its octal equivalent. 11 5 A 8 90 r = 2 ← LSB 16 16 1 0 1 16 1 8 11 r =3
0 80 + 10 = 9010 8 1 r = 1 ← MSB
∴ 5AH = 1328 19 Example: Octal → Binary Example: Octal → Binary Example: Convert the octal number 1328 into its binary equivalent. Convert the octal number 132 into its binary equivalent Solution: First convert the octal number into its decimal equivalent, then convert the d decimal number into its binary equivalent. l b b l 45 1 3 2 2 90 r = 0 ← LSB 22 82 81 80 2 45 r = 1 11 2 22 r = 0 1328 = 10110102 64 8 1 5 2 11 r = 1 2 64 + 24 + 2 = 9010 2 5 r =1 1 2 2 r =0 0 20 2 1 r = 1 ← MSB Binary ↔ Octal ↔ Hex Shortcut Binary ↔ Octal ↔ Hex Shortcut • Relation • Binary, octal, and hex number systems Bi l dh b • All powers of two • Exploit (This Relation) •Make conversion easier.
21 Substitution Code Substitution Code Convert 010101101010111001101010 Convert 0101011010101110011010102 to hex using to hex using the 4‐bit substitution code :
5 6 A E 6 A
0101 0110 1010 1110 0110 1010
56AE6A16 Substitution Code Substitution Code
Substitution code can also be used to convert binary to octal by using 3‐bit groupings:
2 5 5 2 7 1 5 2 010 101 101 010 111 001 101 010
255271528 Other Representation Other Representation • Signed & Unsigned Number • Signed number last bit (one MSB) is signed bit Si d b l bi ( MSB) i i d bi Assume: 8 bit number Unsigned 12 : 0000 1100 Signed +12 : 0000 1100 Signed ‐12 : 1000 1100 • Complement number p Unsigned binary 12 = 00001100 1’ss Complement of 12 = 1111 1 Complement of 12 = 1111 0011 24 Thanks
Combined Balanced Ternary Number System: An Approach to a New Computational Number System combining The Ternary Number System and the Balanced Ternary Number System in the field of Computational Mathematics
Combined Balanced Ternary Number System: An Approach to a New Computational Number System combining The Ternary Number System and the Balanced Ternary Number System in the field of Computational Mathematics
