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7. Explain why we really are using base 1000 notation when we break large decimal integers into blocks of three digits, separated by commas.

7. Explain why we really are using base 1000 notation when we break large decimal integers into blocks of three digits, separated by commas.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.7: Introduction To Coding Theory (optional)
Problem 12E: Suppose that the check digit is computed as described in Example . Prove that transposition errors...
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Secton 2.1 Excerciser # 7

11. Show that any weight not exceeding 2k – 1 may be measured using weights of 1, 2, 2²,.. weights of 1, 3, 3?, ..., 3k-1, when the weights may be placed in either pan. We note that a conversion between two different bases is as easy as binary-hex 16. Show that if n = (afa-1. a¡ɑŋ)b, then the quotient and remainder when n is divided by 14. Explain how to convert from base 3 to base 9 notation, and from base 9 to base 3 notation. 13. Use Exercise 12 to show that any weight not exceeding (3* – 1)/2 may be measured using where a #0 and 0 <a; < | b | forj=0, 1, 2, . . . , k. We write n = (akAk–1 · · . A¡ɑq)b» just 50 Integer Representations and Operations We note that a conversion between two different bases is as easy as conversion whenever one of the bases is a power of the other. 2.1 EXERCISES 1. Convert (1999) 10 from decimal to base 7 notation. Convert (6105)7 from base 7 to decimal notation. 2. Convert (89156) 10 from decimal to base 8 notation. Convert (706113)g from base 8 to decimal notation. 3. Convert (10101111)2 from binary to decimal notation and (999)10 from decimal to binary notation. 4. Convert (101001000), from binary to decimal notation and (1984)10 from decimal to binary notation. 5. Convert (100011110101)2 and (11101001110), from binary to hexadecimal. 6. Convert (ABCDEF)16, (DEFACED) 16, and (9AOB) 16 from hexadecimal to binary. 7. Explain why we really are using base 1000 notation when we break large decimal integers into blocks of three digits, separated by commas. 8. Show that if b is a negative integer less than -1, then every nonzero integer n can be uniquely written in the form n = a,b* +ax-1b*-l +.+a¡b+ ao, where a, ±0 and 0<a; < |b| for j =0, 1, 2, . . , k. We write n = (a,a}-1a¡ɑo)b•Jusi as we do for positive bases. %3D 9. Find the decimal representation of (101001)_2 and (12012)_3. 10. Find the base -2 representations of the decimal numbers –7, –17, and 61. - 2k-1, when all the weights are placed in one pan. 12. Show that every nonzero integer can be uniquely represented in the form e34 +ek-135-1+..+e3+ e0. where ej anced ternary expansion. %3D weights of 1, 3, 3“, ..., 3-1, when the weights may be placed in either pan. - when r> 1 and n are positive integers. %3D are q = (a,a-1.aj)b and r= (aj-1 -...aja)p, respectively. ... %3D .3B %3D ...
Transcribed Image Text:11. Show that any weight not exceeding 2k – 1 may be measured using weights of 1, 2, 2²,.. weights of 1, 3, 3?, ..., 3k-1, when the weights may be placed in either pan. We note that a conversion between two different bases is as easy as binary-hex 16. Show that if n = (afa-1. a¡ɑŋ)b, then the quotient and remainder when n is divided by 14. Explain how to convert from base 3 to base 9 notation, and from base 9 to base 3 notation. 13. Use Exercise 12 to show that any weight not exceeding (3* – 1)/2 may be measured using where a #0 and 0 <a; < | b | forj=0, 1, 2, . . . , k. We write n = (akAk–1 · · . A¡ɑq)b» just 50 Integer Representations and Operations We note that a conversion between two different bases is as easy as conversion whenever one of the bases is a power of the other. 2.1 EXERCISES 1. Convert (1999) 10 from decimal to base 7 notation. Convert (6105)7 from base 7 to decimal notation. 2. Convert (89156) 10 from decimal to base 8 notation. Convert (706113)g from base 8 to decimal notation. 3. Convert (10101111)2 from binary to decimal notation and (999)10 from decimal to binary notation. 4. Convert (101001000), from binary to decimal notation and (1984)10 from decimal to binary notation. 5. Convert (100011110101)2 and (11101001110), from binary to hexadecimal. 6. Convert (ABCDEF)16, (DEFACED) 16, and (9AOB) 16 from hexadecimal to binary. 7. Explain why we really are using base 1000 notation when we break large decimal integers into blocks of three digits, separated by commas. 8. Show that if b is a negative integer less than -1, then every nonzero integer n can be uniquely written in the form n = a,b* +ax-1b*-l +.+a¡b+ ao, where a, ±0 and 0<a; < |b| for j =0, 1, 2, . . , k. We write n = (a,a}-1a¡ɑo)b•Jusi as we do for positive bases. %3D 9. Find the decimal representation of (101001)_2 and (12012)_3. 10. Find the base -2 representations of the decimal numbers –7, –17, and 61. - 2k-1, when all the weights are placed in one pan. 12. Show that every nonzero integer can be uniquely represented in the form e34 +ek-135-1+..+e3+ e0. where ej anced ternary expansion. %3D weights of 1, 3, 3“, ..., 3-1, when the weights may be placed in either pan. - when r> 1 and n are positive integers. %3D are q = (a,a-1.aj)b and r= (aj-1 -...aja)p, respectively. ... %3D .3B %3D ...
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