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Contemporary Mathematics- MAT 130

Numeration Systems (Chapter 4)

Bergen Community College Learning Assistance Center Room: L-125 Phone Number: 201-447-7489

1. Write each of the following Hindu Arabic numeral in expanded form:

a) 573

b) 9052

c) 7202

d) 72025

e) 1010102

2. Express each expanded form as Hindu- Arabic numeral.

a) 5 4 3 2 1 01 10 3 10 0 10 3 10 0 10 9 10

b) 7 4 2 07 10 4 10 6 10 9 10

3. Complete the following table

BASE 10

BASE 2

BASE 5

BASE 8

BASE 12

BASE 16

736

10011101

321

7524

A1B

A1B

Note:

Base 12: A=10, B=11 Base 16: A=10 B=11 C=12 D=13 E =14 F=15




4. Complete the following table in the given base.

Addition Base 5 Multiplication Base 5

+ 0 1 2 3 4 X 0 1 2 3 4

0 0

1 1

2 2

3 3

4 4

Multiplication Base 8

X 0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

7




5. Complete the following table of conversions. Give the answer in the base of the rows. i.e.

row 1 in base seven, row 2 in base two, row 3 in base sixteen, etc.

Convert

to base

5seven 110two ABsixteen 24five 276eight 10011two 24twelve 10

Seven

Two

Sixteen

Five

Eight

Twelve

Ten

YOU MAY USE THIS SPACE FOR COMPUTATIONS




6. Perform the indicated operations in the given base.

a) 22 13four four

b) 24 17twelve twelve

c) 34 24five five

d) 276 12eight eight

e) 10011 11100two two

f) sixteen sixteenAB BE

g) 542 4seven seven

h) 1111 11two two

i) 2 6sixteen sixteenD A

j) 3132 2320four four

k) 13 24twelve twelveA

l) 1010 44five five

m) 702 264eight eight

n) 100110 1010two two

o) 156sixteen sixteenAC

p) 3 223four four

q) 4 143five five

r) 11 1101two two

7. Indicate whether the following are true or false:

a) 2 4 mod3

b) 3 4 mod3

c) 15 4 mod11

d) 3 25 mod11

e) 105 100 mod 2

f) 19 9 mod5

g) 29 9 mod10

h) 2 29 mod15

i) 30 25 mod15

j) 11 25 mod5




8. For each problem fill in the blank with the right choice(s) from the given numbers. Some

number may be used more than once.

9

2

11

3

12

16

4

15

25

98

a) 2 4 mod__

b) 6 __ mod3

c) 12 9 mod__

d) 49 9 mod__

e) __ 5 mod11

f) 100 55 mod__

g) 44 55 mod__

h) 5 __ mod3

i) 8 12 mod__

j) 112 __ mod10

9. Construct an addition table for the 12-hr clock

10. Construct an addition table for the 7-hr clock

11. Construct a multiplication table for the 5-hr clock




Answers:

1.

a) 2 1 0573 5 10 7 10 3 10

b) 3 2 1 09052 9 10 0 10 5 10 2 10

c) 3 2 1 07202 7 10 2 10 0 10 2 10

d) 4 3 2 1 072025 7 10 2 10 0 10 2 10 5 10

e) 6 5 4 3 2 1 01010102 1 10 0 10 1 10 0 10 1 10 0 10 2 10

2.

a) 130309

b) 70040609

3.

BASE 10

BASE 2

BASE 5

BASE 8

BASE 12

BASE 16

736 1011100000 10421 1340 514 2E0

157 10011101 1112 235 111 9D

86 1010110 321 126 72 56

3924 111101010100 111144 7524 2330 F54

1463 10110110111 21323 2667 A1B 5B7

2587 101000011011 40322 5033 15B7 A1B




4. Note: Multiplication of different bases is achieved by converting the product of the numbers

to base five.

Ex: 2x3=6; 6 to base five is 11.

Addition Base 5 Multiplication Base 5

+ 0 1 2 3 4 X 0 1 2 3 4

0 0 1 2 3 4 0 0 0 0 0 0

1 1 2 3 4 10 1 0 1 2 3 4

2 2 3 4 10 11 2 0 2 4 11 13

3 3 4 10 11 12 3 0 3 11 14 22

4 4 10 11 12 13 4 0 4 13 22 31

Multiplication Base 8

X 0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7

2 0 2 4 6 10 12 14 16

3 0 3 6 11 14 17 22 25

4 0 4 10 14 20 24 30 34

5 0 5 12 17 24 31 36 43

6 0 6 14 22 30 36 44 52

7 0 7 16 25 34 43 52 61




5.

The conversion of all the numbers to all of the given bases is:

Convert

to base

5seven 110two ABsixteen 24five 276eight 10011two 24twelve 10

Seven 5 6 333 20 361 25 40 13

Two 101 110 10101011 1110 10111110 10011 11100 1010

Sixteen 5 6 AB E BE 13 1C A

Five 10 11 1141 24 1230 34 103 20

Eight 5 6 253 16 276 23 34 12

Twelve 5 6 123 12 13A 17 24 A

Ten 5 6 171 14 190 19 28 10




6. Note: r stands for remainder

a) 101four

b) 3 twelveB

c) 113 five

d) 310eight

e) 101111two

f) 169sixteen

g) 3131seven

h) 101101two

i) 57 4sixteenF

j) 212 four

k) 116twelve

l) 411five

m) 416eight

n) 11100two

o) sixteenAA

p) 32 1 four fourr

q) 22 five

r) 100 1 two twor




7.

a) F

b) F

c) T

d) T

e) F

f) T

g) T

h) F

i) F

j) F

8.

a) 2

b) 3, 9, 12, 15

c) 3

d) 2, 4

e) 16

f) 3, 9, 15

g) 11

h) 2, 11, 98

i) 2, 4

j) 2, 12

9.

+ 0 1 2 3 4 5 6 7 8 9 10 11

0 0 1 2 3 4 5 6 7 8 9 10 11

1 1 2 3 4 5 6 7 8 9 10 11 0

2 2 3 4 5 6 7 8 9 10 11 0 1

3 3 4 5 6 7 8 9 10 11 0 1 2

4 4 5 6 7 8 9 10 11 0 1 2 3

5 5 6 7 8 9 10 11 0 1 2 3 4

6 6 7 8 9 10 11 0 1 2 3 4 5

7 7 8 9 10 11 0 1 2 3 4 5 6

8 8 9 10 11 0 1 2 3 4 5 6 7

9 9 10 11 0 1 2 3 4 5 6 7 8

10 10 11 0 1 2 3 4 5 6 7 8 9

11 11 0 1 2 3 4 5 6 7 8 9 10




10.

+ 0 1 2 3 4 5 6

0 0 1 2 3 4 5 6

1 1 2 3 4 5 6 0

2 2 3 4 5 6 0 1

3 3 4 5 6 0 1 2

4 4 5 6 0 1 2 3

5 5 6 0 1 2 3 4

6 6 0 1 2 3 4 5

11.

X 0 1 2 3 4

0 0 0 0 0 0

1 0 1 2 3 4

2 0 2 4 1 3

3 0 3 1 4 2

4 0 4 3 2 1