Binary numbersystem1
Download as PPT, PDF0 likes833 views
The document discusses binary number conversion between decimal and binary representations. It provides two processes: 1) Decimal to binary conversion uses successive division, where the decimal number is divided by 2 and the remainders form the binary number bits. 2) Binary to decimal conversion uses weighted multiplication, where each bit of the binary number is multiplied by its place value and the products are summed to obtain the decimal number. Worked examples demonstrate converting specific values between decimal and binary.
Binary numbersystem1
- 1. Binary Number System And Conversion
- 2. Bridging the Digital Divide 16 2 3 3 64 721 93355 3 16 63 2 Decimal-to-Binary 3 721 53 935 Conversion 5 5 4 234 1257137 2 9 935 13 75 4 34 7 145 2 01 01 0010 01 01 10011 1 100010110 01 0 0 010 1101 010 1 1 01011 01 1 1100 001 0 10 11 0 001 011 00 010 01 0 1 01 01 101 1 1 011011 01 110 001010010 Binary-to-Decimal Conversion 101 101 0 1 11 01 01010 11 0 0101011 1001 0 0 1 01 101 1001 011101 1 0 00 1 001011 00 1 1 00 2
- 3. Decimal ‒to‒ Binary Conversion The Process : Successive Division a) Divide the Decimal Number by 2; the remainder is the LSB of Binary Number . b) If the quotation is zero, the conversion is complete; else repeat step (a) using the quotation as the Decimal Number. The new remainder is the next most significant bit of the Binary Number. Example: Convert the decimal number 610 into its binary equivalent. 3 2 6 r = 0 ← Least Significant Bit 1 2 3 r =1 ∴ 610 = 1102 0 2 1 r = 1 ← Most Significant Bit 3
- 4. Dec → Binary : Example #1 Example: Convert the decimal number 2610 into its binary equivalent. 4
- 5. Dec → Binary : Example #1 Example: Convert the decimal number 2610 into its binary equivalent. Solution: 13 2 26 r = 0 ← LSB 6 2 13 r =1 3 2 6 r=0 ∴ 2610 = 110102 1 2 3 r =1 0 2 1 r = 1 ← MSB 5
- 6. Dec → Binary : Example #2 Example: Convert the decimal number 4110 into its binary equivalent. 6
- 7. Dec → Binary : Example #2 Example: Convert the decimal number 4110 into its binary equivalent. Solution: 20 2 41 r = 1 ← LSB 10 2 20 r=0 5 2 10 r=0 ∴ 4110 = 1010012 2 2 5 r =1 1 2 2 r=0 0 2 1 r = 1 ← MSB 7
- 8. Dec → Binary : More Examples a) 1310 = ? b) 2210 = ? c) 4310 = ? d) 15810 = ? 8
- 9. Dec → Binary : More Examples a) 1310 = ? 11012 b) 2210 = ? 101102 c) 4310 = ? 1010112 d) 15810 = ? 100111102 9
- 10. Binary ‒to‒ Decimal Process The Process : Weighted Multiplication a) Multiply each bit of the Binary Number by it corresponding bit- weighting factor (i.e. Bit-0→20=1; Bit-1→21=2; Bit-2→22=4; etc). b) Sum up all the products in step (a) to get the Decimal Number. Example: Convert the decimal number 01102 into its decimal equivalent. 0 1 1 0 23 22 21 20 Bit-Weighting ∴ 0110 2 = 6 10 8 4 2 1 Factors 0 + 4 + 2 + 0 = 610 10
- 11. Binary → Dec : Example #1 Example: Convert the binary number 100102 into its decimal equivalent. 11
- 12. Binary → Dec : Example #1 Example: Convert the binary number 100102 into its decimal equivalent. Solution: 1 0 0 1 0 24 23 22 21 20 16 8 4 2 1 16 + 0 + 0 + 2 + 0 = 1810 ∴100102 = 1810 12
- 13. Binary → Dec : Example #2 Example: Convert the binary number 01101012 into its decimal equivalent. 13
- 14. Binary → Dec : Example #2 Example: Convert the binary number 01101012 into its decimal equivalent. Solution: 0 1 1 0 1 0 1 26 25 24 23 22 21 20 64 32 16 8 4 2 1 0 + 32 + 16 + 0 + 4 + 0 + 1 = 5310 ∴01101012 = 5310 14
- 15. Binary → Dec : More Examples a) 0110 2 = ? b) 11010 2 = ? c) 0110101 2 = ? d) 11010011 2 = ? 15
- 16. Binary → Dec : More Examples a) 0110 2 = ? 6 10 b) 11010 2 = ? 26 10 c) 0110101 2 = ? 53 10 d) 11010011 2 = ? 211 10 16
- 17. Summary & Review Successive Division a) Divide the Decimal Number by 2; the remainder is the LSB of Binary Number . b) If the Quotient Zero, the conversion is complete; else repeat step (a) using the Quotient as the Decimal Number. The new remainder is the next most significant bit of the Binary Number. Weighted Multiplication a) Multiply each bit of the Binary Number by it corresponding bit-weighting factor (i.e. Bit-0→20=1; Bit-1→21=2; Bit-2→22=4; etc). b) Sum up all the products in step (a) to get the Decimal Number. 17
Editor's Notes
- #2: Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #3: Introductory Slide / Overview of Presentation Explain that humans use base ten (or decimal), because we have ten fingers and that digital electronics uses base-two (binary) because it only understands two states; ON and OFF. For students to be able to analyze and design digital electronics, they need to be proficient at converting numbers between these two number systems. Base ten has ten unique symbols (0 – 9) while binary has two unique symbols (0 – 1). Any number can represent a base and the number of symbols it utilizes will always be that number. This is discussed further later in Unit 2. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #4: Review the DECIMAL-to-BINARY conversion process. Remind the students to subscript all numbers (i.e. Subscript 10 for decimal & subscript 2 for binary) A common mistake is inverting the LSB and MSB. The three-dot triangular symbol here stands for the word “therefore” and is used commonly among mathematics scholars. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #5: Pause the power point and allow the student to work on the example. The solution is on the next slide. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #6: Here is the solution. If you print handouts, don’t print this page. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #7: Pause the power point and allow the student to work on the example. The solution is on the next slide. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #8: Here is the solution. If you print handouts, don’t print this page. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #9: If the students need more practice, here are four additional example of DECIMAL to BINARY conversion. The solution is on the next slide. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #10: Here are the solutions. If you print handouts, don’t print this page. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #11: Review the BINARY-to-DECIMAL conversion process. Remind the students to subscript all numbers (i.e. Subscript 10 for decimal & subscript 2 for decimal) Let the students know that as the become more proficient at the conversions, they may not need to write out the Bit-Weighting Factors. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #12: Pause the power point and allow the student to work on the example. The solution is on the next slide. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #13: Here is the solution. If you print handouts, don’t print this page. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #14: Pause the power point and allow the student to work on the example. The solution is on the next slide. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #15: Here is the solution. If you print handouts, don’t print this page. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #16: If the students need more practice, here are four additional example of DECIMAL to BINARY conversions. The solution is on the next slide. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #17: Here are the solutions. If you print handouts, don’t print this page. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009
- #18: Prior to assigning the activity, review the process for DECIMAL-to-BINARY and BINARY-to-DECIMAL. Binary Number System Digital Electronics 2.1 Introduction to AOI Logic Project Lead The Way, Inc. Copyright 2009