Base 10 and Other Bases (Exercises with Answers)

Base 10 and other bases Possible Digits 0, 1 0, 1, 2, 3, 4, 5 0, 1, 2, 3, 4, 5, 6 0, 1, 2, 3, 4, 5, 6, 7 0, 1, 2, 3, 4, 5, 6, 7, 8,9 A, B, C, D, E, F, 0, 1, 2 0, 1, 2, 3 0, 1, 2 , 3, 4 Base Four Base Five Base Six Base Seven Base Eight Base 16 Base Ten Base Ten Base Two Base Three 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 10 2 2 2 2 2 2 2 2 3 11 10 3 3 3 3 3 3 3 4 100 10 4 4 4 4 4 4 5 101 10 5 5 5 5 5 10 6 6 6 6 10 7 7 7 10 8 8 9 9 10 A 10 11 B 11 C 12 D 13 E 14 F 15 10 16 11 17 12 18 13 19 14 20 15 21 22 16 22 23 17 23 18 24 19 25 1A 26 6 7 8 1000 9 12 100 1100 20 13 14 20 15 16 30 10000 100 20 17 18 200 30 19 103 20 110 40 21 30 24 11000 25 11001 40 221 30 100 26 27 27 123 28 28 40 29 11101 30 11110 29 1010 132 110 50 42 36 1E 30 1 Base 10 and other bases 0, 1, 2, 3, 4, 5 0, 1, 2, 3, 4, 5, 6 0, 1, 2, 3, 4, 5, 6, 7 0, 1, 2, 3, 4, 5, 6, 7, 8,9 A, B, C, D, E, F, Base Ten Base Two 0, 1, 2 Base Three 0, 1, 2, 3 0, 1, 2 , 3, 4 Base Four Base Five Base Six Base Seven Base Eight Base 16 Base Ten 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 10 2 2 2 2 2 2 2 2 3 11 10 3 3 3 3 3 3 3 4 100 11 10 4 4 4 4 4 4 5 101 12 11 10 5 5 5 5 5 6 110 20 12 11 10 6 6 6 6 7 111 21 13 12 11 10 7 7 7 8 1000 22 20 13 12 11 10 8 8 9 1001 100 21 14 13 12 11 9 9 10 1010 101 22 20 14 13 12 A 10 11 1011 102 23 21 15 14 13 B 11 12 1100 110 30 22 20 15 14 C 12 13 1101 111 31 23 21 16 15 D 13 14 1110 112 32 24 22 20 16 E 14 15 1111 120 33 30 23 21 17 F 15 16 10000 121 100 31 24 22 20 10 16 17 10001 122 101 32 25 23 21 11 17 18 10010 200 102 33 30 24 22 12 18 19 10011 201 103 34 31 25 23 13 19 20 10100 202 110 40 32 26 24 14 20 21 10101 210 111 41 33 30 25 15 21 22 10110 211 112 42 34 31 26 16 22 23 10111 212 113 43 35 32 27 17 23 24 11000 220 120 44 40 33 30 18 24 25 11001 221 121 100 41 34 31 19 25 26 11010 222 122 101 42 35 32 1A 26 27 11011 1000 123 102 43 36 33 1B 27 28 11100 1001 130 103 44 40 34 1C 28 29 11101 1002 131 104 45 41 35 1D 29 30 11110 1010 132 110 50 42 36 1E 30 Possible Digits 0, 1 2 Practice # Date:_________ Section: _______ Name: ____________________________ ______ Find the next five numbers before and after Before Base 5 (0,1,2,34) After Before Base 4 (0,1,2,3) (4220)5 (4214)5 After (3323) 4 (3322) 4 Find the next five numbers before and after Before Base 2 (0,1) After Before (1111) 2 (1110) 2 Base 3 (0,1,2) After (1000)3 (222)3 3 Practice # Date:_________ Section: _______ Name: ____________________________ ______ Find the next five numbers before and after Before After Before After (4204)5 (4220)5 (3311) 4 (3323) 4 (4210)5 (4221)5 (3312) 4 (3330) 4 (4211)5 (4222)5 (3313) 4 (3331) 4 (4212)5 (4223)5 (3320) 4 (3332) 4 (4213)5 (4224)5 (3321) 4 (3333) 4 (4214)5 (4230)5 (3322) 4 (10000) 4 Find the next five numbers before and after Before After Before After (1001) 2 (1111) 2 (210)3 (1000)3 (1010) 2 (10001) 2 (211)3 (1001)3 (1011) 2 (10010) 2 (212)3 (1002)3 (1100) 2 (10011) 2 (220)3 (1010)3 (1101) 2 (10100) 2 (221)3 (1011)3 (1110) 2 (10101) 2 (222)3 (1012)3 4 Conversion from other bases to base 10 Hint: we use expansion with given base. 1) Convert (213) 4 to base 10. Step 1) Write the number in expanded form 2(42 ) + 1(41 ) + 3(40 ) Step 2) 2(16) + 1(4) + 3(1) = 32 + 4 + 3 = 39 2) Convert (10011) 2 to base 10. Step 1) Write the number in expanded form 1(24 ) + 0(23 ) + 0(42 ) + 1(21 ) + 1(21 ) Step 2) 1(16) + 0(8) + 0(4) + 1(2) + 1(1) = 16 + 0 + 0 + 2 + 1 = 19 3) Convert (234)5 to base 10. Step 1) Write the number in expanded form 2(52 ) + 3(51 ) + 4(50 ) Step 2) 2(25) + 3(5) + 4(1) = 50 + 15 + 4 = 69 4) Convert ( ABE )16 to base 10. Step 1) Write the number in expanded form A(162 ) + B(161 ) + E (160 ) = 10(256) + 11(16) + 14(1) = 2750 Step 2) A(256) + B(16) + E (1) 5 Base Conversion Date:_________ Section: _______ Name: _________________________________ 1. Convert 1010101two to base ten 1. 85 2. Convert 404 five to base ten 2. 104 3. Convert 4205six to base ten 3. 941 4. Convert 1221three to base ten 4. 52 5 Convert 3B 2 sixteen to base ten 5. 946 6. Convert 1010111two to base ten 6. 87 7. Convert 432 five to base ten 7. 117 8. Convert 1542 six to base ten 8. 422 9. Convert 12121three to base ten 9. 41 10 Convert ABEsixteen to base ten 10. 2750 11 Convert BADsixteen to base ten 11. 2989 12 Convert 2 FADsixteen to base ten 12. 12205 6 Conversion from base 10 to other Bases Important: The answer is the last quotient and remainder numbers from left to right. 5) Convert 39 to base Four Step 1 Step 2 Divide by 4 Quotient Remainder 9 3 39 ÷ 4 2 1 9÷4 Stop because quotient is smaller than base Answer (213) 4 6) Convert 19 to base Two Step 1 Step 2 Step 3 Step 4 Divide by 2 Quotient Remainder 9 1 19 ÷ 2 4 1 9÷2 2 0 4÷2 1 0 2÷2 Stop because quotient is smaller than base Answer (10011) 2 7) Convert 500 to base Five Step 1 Step 2 Step 3 Divide by 5 Quotient Remainder 100 0 500 ÷ 5 20 0 100 ÷ 5 4 0 20 ÷ 5 Stop because quotient is smaller than base Answer (4000)5 8) Convert 2750 to base Sixteen Important, You need to know that in base sixteen, 10=A, 11=B, 12=C, 13=D, Step 1 Step 2 14=E, 15=F Divide by 16 Quotient Remainder 2750 ÷ 16 14 14=E 171 ÷ 16 10=A 11=B Stop because quotient is smaller than base Answer ( ABE )16 7 PQ3 # ____ Date:_________ Section: _______ Name: _____________________________________ Convert each of the following from base ten to the indicated base. 1. Convert 53 to base Two 1. 110101 2. Convert 63 to base Three 2. 2100 3. Convert 153 to base Eight 3. 231 4. Convert 95 to base Five 4. 340 5. Convert 2603 to base Sixteen 5. A2B 6. Convert 75 to base Four 6. 1023 7. Convert 87 to base Three 7. 10020 8. Convert 501 to base Eight 8. 765 9. Convert 115 to base Five 9. 430 10. Convert 486 to base Sixteen 10. 1E6 8 Practice on Base conversion For each problem: first convert to base 10 and then to the given base Base 10 1 Given Base ( 234 ) 6 Base 2 Answer:94 2 3 4 5 6 ( 536 ) 7 (1111011) 2 ( CAB )16 ( 3333) 5 ( 443)6 Answer: 1011110 Base 9 Answer:332 Answer:272 Base 16 Answer:7B Answer:123 Base 8 Answer:6253 Answer:3243 Base 7 Answer:468 Answer:171 Answer:1236 Base 16 Answer:AB Find the digit in the given base on the right side to be equal to the number given the right with the given base 1) 2) 3) ( 5 x6 ) 7 = ( 2042 ) 5 (11x 21) 3 = ( 311) 6 ( 3x 2 ) 7 = ( AA) 16 Answer: x = 3 Answer: x = 0 Answer: x = 5 9 Practice: Convert from one base to another base Date:_________ Section: _______ Name: ______________________ 1 Convert 505six to base eight 1) 271 eight 2. Convert 2210three to base five 2) 300 five 3 Convert 6253eight to base sixteen 3) CAB sixteen 4. Convert 1234 five to base sixteen 4) C2 sixteen 5 Convert 2 ACsixteen to base nine 5) 840 nine 6. Convert 1010001two to base six 6) 213 six 7. Convert 432 five to base four 7) 1311 four 8. Convert 3205six to base sixteen 8) 2D5 sixteen 9. Convert 1211three to base nine 9) 10 Convert 2 A3sixteen to base five 10) 10200 five 54 nine 10 Practice Add the numbers with the same bases Date:________ 1) 342 five + 404 five = Section: _______ Name: __________________ 1) 1301 five 2) 342 six + 404 six = 2) 1150 six 3) 4205six + 3345six = 3) 11554 six 4) 4205eight + 3345eight = 4) 7552eight 5) 10101two + 11101two = 5) 1100110two 6) 1010111two + 1110101two = 6) 11001100two 7) BBCsixten + ABCsixteen = 7) 1678sixteen 8) BADsixten + ABEsixteen = 8) 166 Bsixteen 9) ABE1sixten + DAD 2 sixteen = 9) 186 B3sixteen 10) ABCDsixten + BABAsixteen = 10) 16687 sixteen 11 Practice Subtract the numbers with the same bases Date:________ 1) 3213 four − 1122 four = Section: _______ Name: _______________ 1) 2031 four 2) 3213 five − 1122 five = 2) 2041 five 3) 403 five − 304 five = 3) 44 five 4) 4205eight − 3346eight = 4) 637 eight 5) 11010two − 10101two = 5) 101two 6) 10000two − 1two = 6) 1111two 7) 54321six − 12345six = 7) 41532 six 8) BADsixten − ABEsixteen = 8) EFsixteen 9) FACEsixten − DEADsixteen = 9) 1C 21sixteen 10) DEADsixten − BEEFsixteen = 10) 1FBEsixteen 12