chapter 18 i number bases enhance

8/3/2019 Chapter 18 I Number Bases ENHANCE

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CHAPTER 18: NUMBER BASES

1.1 Important concepts

1.2 Number in base two

1.2.1 State the value of a digit of a number in base two.

Example: 11012

The value of a digit 1 is 1 × 23 = 8

Exercises :

1. 11002 2. 1101102

3. 1000102 4. 11101002

5. 1001002 6. 10111112

7. 101111102 8. 111101102

9. 101011002 10. 10001111002

1.2.2 Conversion of numbers in base two to the number in base ten.

Example : 1102 = (1 × 22 ) + (1 × 21 ) + (0 × 20 )

= 4 + 2 + 0

Number Bases 12

Number Bases

Number in base

two

Only use digits 0

and 1.

Place value :

23222120

Number in base

eight

Only use digits 0,

1, 2, 3, 4, 5, 6 and

7,

Place value :

83828180

Number in base

five

Only use digits 0,

1, 2, 3 and 4

Place value :

53525150



= 6 10

Exercises :

1. 1012 2. 11012

3. 11002 4. 11102

5. 10112 6. 11012

7. 101102 8. 110112

Example : 10012 =

= 8 + 1

= 910

9. 1011102 10. 1110002

11

.

11001112 12. 10100112

13

.

101011012 14. 111011012

1.2.3 Conversion of numbers in base ten to the number in base two.

Example : 2510 2 25 remainder

2 12 1

2 6 0

2 3 0

2 1 1

0 1

2510 = 110012

Exercises :

1. 1610 2. 2910

Number Bases 13

8 4 2 1

1 0 0 1



3. 3410 4. 3910

5. 4410 6. 4710

Example : 1210

1210 = 11002

7. 5510 8. 6810

9. 8710 10. 9410

11

.

10110 12. 11710

1.3 Number in base eight

1.3.1 State the value of a digit of a number in base eight.

Example : 142


Exercises :

1. 158 2. 248

3. 618 4. 1458

5. 3278 6. 3268

7. 25168 8. 34048

9. 16258 10. 46138

1.3.2 Conversion of numbers in base eight to the number in base ten.

Example: 26 8 = ( 2 × 81 ) + (6 × 80 )

= 16 + 6 = 2210

Exercises:

1. 238 2. 348

3. 638 4. 718

5. 1248 6. 1568

Number Bases 14

4 2 1

1 1 0 0

8 4



7. 2708 8. 3748

Example : 458 =

= 32 + 5

= 37

9. 2058 10. 2728

11

.

3168 12. 3658

13

.

5138 14. 6528

1.3.3 Conversion of numbers in base ten to the number in base eight.

Example: 1910 8 19 remainder

8 2 3

0 2

1910 = 238

Exercises :

1. 2610 2. 4910

3. 6410 4. 9910

5. 14410 6. 54710

Example : 17 10

17 10 = 218

7. 1910 8. 2410

9. 5710 10. 6410

Number Bases 15

512 64

2 1

8 1

51

2

6

4

8 1

4 5



11

.

12110 12. 21710

1.4 Number in base five

1.4.1 State the value of a digit of a number in base five.

Example : 225


Exercises :

1. 145 2. 235

3. 325 4. 445

5. 1015 6. 1245

7. 2405 8. 12125

9. 32045 10. 42215

1.4.2 Conversion of numbers in base five to the number in base ten.

Example: 1425 = ( 1 × 52 ) + ( 4 × 51 ) + ( 2 × 50 )

= 25 + 20 + 2

= 47 10

Exercises :

1. 135 2. 205

3. 315 4. 425

5. 2115 6. 3025

7. 31105 8. 43215

Example : 1245 =

= 25 + 10 + 4

= 3910

9. 125 10. 215

Number Bases 16

125 25 5 1

1 2 4



11

.

1045 12. 2305

13

.

22345 14. 42135

1.4.3 Conversion of numbers in base ten to the number in base five.

Example : 2810 5 28 remainder

5 5 3

5 1 0

0 1

2810 = 1035

Exercises :

1. 1610 2. 2910

3. 7710 4. 8510

5. 14410 6. 44710

Example : 1210

1210 = 225

7. 5610 8. 7810

9. 8710 10. 10410

11

.

20110 12. 31710

1.5 Conversion of number in certain base to a number in another base.

1.5.1 Conversion of number in base two to the number in base eight.

Example : 111012

4 2 1 4 2 1 4 2 1

1 1 1 0 1

3 5111012 = 358

Number Bases 17

12

5

2

5

2 2

5 1



1. 11012 2. 10012

3. 110012 4. 110112

5. 1001112 6. 1110102

7. 11011112 8. 11010002

9. 100111012 10 111101102

11 1001110112 12 10011101102

13 100111011012 14 1101010100112

1.5.2 Conversion of number in base eight to the number in base two.

Example: 27 8

2 7 4 2 1 4 2 1 4 2 1

0 1 0 1 1 127 8 = 101112

1. 258 2. 368

3. 518 4. 2028

5. 5368 6. 6438

7. 12358 8. 25718

9. 32248 10 60478

11 25508 12 51018

13 710018 14 562218

Number Bases 18



1.5.3 Conversion of number in base five to the number in base two.

Example : 145

145= 5 + 4= 910

145 = 910 = 10012

Exercises :

1. 235 2. 435

3. 1345 4. 2125

5. 3015 6. 4235

1.5.4 Conversion of number in base five to the number in base eight.

Example : 145

145 = 5 + 4 145 = 910 = 118

= 910

Number Bases 19

125 25 5 1

1 4

2 9 remainder

2 4 1

2 2 0

2 1 00 1

125 25 5 1

1 4

8 9 remainder

8 1 1

0 1



Exercises :

1. 135 2. 335

3. 1245 4. 2215

5. 3105 6. 4135

1.5.5 Conversion of number in base two to the number in base five.

Example : 1112

1112 = 4 + 2 + 1 7 10 = 125

= 7 10

Exercises :

1. 1012 2. 11102

3. 110112 4. 101102

5. 1110112 6. 1011112

1.5.6 Conversion of number in base eight to the number in base five.

Example : 16 8

Number Bases 20

8 4 2 1

1 1 1

5 7 remainder

5 1 2

0 1

512 64 8 1

1 6

5 14 remainder

5 2 4

0 2



16 8 = 8 + 6 1410 = 245

= 1410

Exercises :

1. 178 2. 338

3. 1748 4. 2218

5. 5108 6. 7138

1.5.7 Perform computations involving addition of two numbers in base two.

Example : 1 0 0 12

+ 1 0 12

1 1 1 02

Exercises :

1. 1012 + 1102 2. 1112 + 11012

3. 101002 + 110112 4. 101102 + 11012

5. 1110112 + 101012 6. 1011112 + 101002

1.5.8 Perform computations involving subtraction of two numbers in base two.

Example : 1 0 0 12

- 1 0 12

1 0 02

Exercises :

Number Bases 21



1. 1012 - 102 2. 1112 - 1012

3. 101002 - 10112 4. 111102 - 101012

5. 1110112 - 1101012 6. 1101002 - 1011112

Objective Questions.

1. State the value of underlined digit in the in the

number .01011102

A A4

B 8

C 16

D 32

2. State the value of underlined digit in the

number .240315

A A75

B 375

C 1875

D 3

3. Express 2110111 as a number in base

ten.

A A1310

B 3110

C 5510

D 11010

4. Express 1548 as a number in base ten.A A8010

B 10810

C 86410

D 192510


ten.

8. Express 93210 as number in base

five.

A 1435

B 2125

C 22125

D 122125


eight.

A A268

B 328

C 1228

D 11018

10. Express 10 110 1012 as a number in base

eight.

A 2358

B 2648

C 2658

D 5518

11 Express 4028 as a number in base two.

A A1100102

B 1100100102

C 10000102

D 1000000102

12 Express 101102 as a number in base five.

Number Bases 22



A A84610

B 42310

C 11310

D 76510

6. Express 12610 as a number in base two.

A A1111102

B 11101102

C 11111102

D 11111112


eight.

A A5448

B 4288

C 27428

D 65448

155Express 7( 82 ) + 5 as a number in base two.

A A111000112

B 110001012

C 11101012

D 1110001012

16 Express 27 + 22 + 1 as a number in base

eight.

A A198

B 1338

C 2058

D 7208

17 Express 10 110 1012 as a number in base

eight.

A 2358

B 2648

C 2658

D 5518

18 Express 53 + 4 as a number in base five.

A 1045

B 3045

C 10045

D 30045

19 If y5 = 110 1112 , then y =

A 205

B 515

C 1605

D 2105

A 225

B 425

C 445

D 40445

13 Express 4135 as a number in base two.

A A10010112

B 100001011

C 1011002

D 11011002

14 Express 5468 as a number in base five.

A A3345

B 4135

C 24135

D 3415

22

Given that 8521 011 11 1 ++= x

, find

the value of x.

A A15

B 16

C 17

D 18

23 If 035

2222 ++= x , then x =

A A101 0002

B 101 0012

C 110 0012

D 110 0112

24. 100102 + 1112 =

A A111012

B 110012

C 101012

D 100012

25. Given that x2 = 1012 + 1015 , find the

value of x.

A 11112

B 10 1112

C 11 1112

D 101 1112

Number Bases 23



20 If 20008 = x 10 which is x is integer, find

the value of x.

A A1600

B 2500

C 1024D 8192

21. If x5 = 758 , then x =

A 61

B 211

C 221

D 111 101

PAST YEAR QUESTIONS

SPM 2003(Nov)

1. Express 11010101 2 as a number in base eight

A. 651 8 C. 8325

B. 8625 D. 8

221

2. =+22

111011101

A. 2100011 C. 2

101111

B. 2

111001

D. 2

101011

SPM 2004(Nov)

3. Express 353+ as a number in base five

A. 5103 C. 5

1003

B. 5303 D. 5

3003

4. =+22

11111011

A. 2100010 C. 2

110011

B. 2101100 D. 2111101

SPM 2004(Jun)

5. What is the value of the digit 3, in base ten, in the number 3728?

A. 48 C. 192

B. 64 D. 512

6. Convert 22 + 2(23 + 1) to a number in base eight

Number Bases 24



A. 158 C. 228

C. 268 D. 628

7. 110112 – 1012 =

A. 10102 C. 101102

B. 11102 D. 100102

SPM 2005(Nov)

8. =−22

1011110001

A. 2100100 C. 2

111010

B. 2100110 D. 2

111100

9. What is the value of the digit 2, in base ten, in the number 53241 ?

A. 25 C. 200B. 50 D. 250

SPM 2005(Jun)

10. 100102 - 11012 =

A. 1012 C. 10112

B. 1112 D. 11112

11. Express 11101002 as a number in base eight

A. 1218 C. 4618

B 1648 D 7208

SPM 2006(Nov)

12. What is the value of the digit 2 in base ten, in the number 1203 5

A. 10 C. 200

B. 50 D. 250

13. =−22

10111001

A. 21001 C. 2

1101

B. 210000 D. 2

10100

SPM 2006(Jun)

14. It is given that k 10 = 1100012, where k is an integer. Find the value of k .

A. 48 C. 97

B. 49 D. 98

15. 1012 + 10012 =A. 11102 C. 100002

Number Bases 25



B. 11002 D. 100112

SPM 2007(Jun)

16. Express 7 x 83 + 2 x 82 + 8 as a number in base eight.

A. 127 8 C. 721 8

B. 3270 8 D. 7210 8

17. 11012 + 1111012 =

A. 1100002 C. 1110102

B. 10010102 D. 1110002

SPM 2007(Nov)

18. 1100102 - 111012 =A. 1011012 C. 1001012

B. 110002 D. 101012

19. Given 3 x 53 + 4 x 52 + 5y = 3420 5 . Find the value of y.

A. 0 C. 3

B 2 D 4

SPM 2008(Jun)

20. Convert 324 8 to a number in base two.

A. 100001011 2 C. 11001100 2

B. 1000100112

D. 110101002

21. 101002 - 11112 =

A. 1012 C. 10012

B. 1112 D. 10112

SPM 2008(Nov)

22. 101012 - 112 =

A. 100002 C. 101002

B. 100102 D. 101102

23. Convert 204 8 to a number in base five.

A. 1012 5 C. 1304 5

Number Bases 26



B. 2101 5 D. 4031 5

ANSWERSChapter 18 Number Bases

1.2.1

1. 4 2. 16 3. 0 4. 64 5. 4

6. 16 7. 32 8. 16 9. 0 10. 32

1.2.21. 5 2. 13 3. 12 4. 14 5. 11

6. 13 7. 22 8. 27 9. 46 10. 56

11. 103 12. 83 13. 173 14. 237

1.2.3

1. 10000 2. 11101 3. 100010 4. 100111 5. 101100

6. 101111 7. 110111 8. 1000100 9. 1010111 10. 1011110

11. 1100101 12. 1110101

1.3.11. 5 2. 4 3. 1 4. 32 5. 16

6. 16 7. 320 8. 512 9. 512 10. 2048

1.3.2

1. 19 2. 28 3. 51 4. 57 5. 84

6. 110 7. 184 8. 252 9. 133 10. 186

11. 206 12. 245 13. 331 14. 426

1.3.3

1. 32 2. 61 3. 100 4. 143 5. 220

6. 1043 7. 23 8. 30 9. 71 10. 100

11. 171 12. 331

1.4.1

1. 4 2. 3 3. 15 4. 20 5. 25

6. 10 7. 0 8. 50 9. 375 10. 500

1.4.2

1. 8 2. 10 3. 16 4. 22 5. 56

6. 77 7. 405 8. 586 9. 7 10. 11

Number Bases 27



11. 29 12. 65 13. 219 14. 558

1.4.3

1. 31 2. 104 3. 302 4. 320 5. 1034

6. 3242 7. 211 8. 303 9. 322 10. 40411. 1031 12. 2232

1.5.1

1. 15 2. 11 3. 31 4. 33 5. 47

6. 72 7. 157 8. 150 9. 235 10. 366

11. 473 12 1166 13. 2355 14. 6523

1.5.2

1. 10101 2. 11110 3. 101001

4. 10000010 5. 101011110 6. 110100011

7. 1010011101 8. 10101111001 9. 11010010100

10. 110000100111 11. 10010010000 12. 10100100000113. 111001000000001 14. 101110010010001

1.5.3

1. 1101 2. 10111 3. 101100 4. 111001 5. 10011006. 1110001

1.5.4

1. 1000 2. 10010 3. 100111 4. 111101 5. 1010000

6. 1101100

1.5.5

1. 10 2. 24 3. 102 4. 42 5 2146. 142

1.5.6

1. 30 2. 102 3. 444 4. 1040 5. 2303

6. 3314

1.5.7

1. 1011 2. 10100 3. 101111 4. 100011 5. 1010000

6. 1000011

1.5.8

1. 11 2. 10 3. 1001 4. 1001 5. 1106. 101

Objective questions

1. C 2. B 3. C 4. B 5. C

6. C 7. D 8. D 9. B 10. C

11. D 12. B 13. D 14. C 15. D

16. C 17. C 18. C 19. D 20. C

21. C 22. D 23. B 24. B 25. C

Number Bases 28



SPM QUESTIONS

1. C 2. D 3. C 4. A 5. C 6. B7. C 8. B 9. B 10. A 11. B 12. B

13. A 14. B 15. A 16. D 17. C 18. D

19. A 20. B 21. A 22. B 23. A

N b B 29

chapter 18 i number bases enhance

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