F4 C2

Chapter 2: Number bases

Number bases
1 Number bases are number systems of consisting of digits from 0 to 9.

325 read as “three two base five”

Number base Digit 0, 1, 10, 11, 100, 101, 110, 111, 1000,
Base 2 0, 1 1001, 1010, 1011, 1100, …
Base 3 0, 1, 2
Base 4 0, 1, 2, 3 0, 1, 2, 10, 11, 12, 20, 21, 22, 100,
Base 5 0, 1, 2, 3, 4 101, …

Base 6 0, 1, 2, 3, 4, 5 0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22,
23, 30, 31, 32, 33, 100, …
Base 7 0, 1, 2, 3, 4, 5, 6
0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21,
Base 8 0, 1, 2, 3, 4, 5, 6, 7
Base 9 0, 1, 2, 3, 4, 5, 6, 7, 8 22, 23, 24, 30, 31, 32, 33, 34, 40, 41,
Base 10 0, 1, 2 3, 4, 5, 6, 7, 8, 9 42, 43, 44, 100, 101, …

0, 1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15,

20, 21, 22, 23, 24, 25, 30, 31, 32, 33,

34, 35, 40, 41, 42, 43, 44, 45, 50, 51,
52, 53, 54, 55, 100, 101, 102, …

0, 1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14,

15, 16, 20, 21, 22, 23, 24, 25, 26, 30,

31, 32, 33, 34, 35, 36, 40, 41, 42, 43,

44, 45, 46, 50, 51, 52, 53, 54, 55, 56,
60, 61, 62, 63, 64, 65, 66, 100, 101, …

12

11
0, 1, 2, 3, 4, … 9, 10, 11, 12, …

1) Write 3 examples of numbers that represent numbers in base two up to base ten.

Number base Number

Base 2 12 102 10012
Base 3 213 223 1003
Base 4 34 334 12014
Base 5 45 105 415
Base 6 56 1006 1016
Base 7 257 347 1007
Base 8 78 128 138
Base 9 89 879 1009
Base 10 3310
64 84

百万 十万 万 千 百 十 个
Million Thousand Hundred Ten Unit
Hundred Ten
106 thousand thousand 103 102 101 100
1 000 000 1 000
105 104 100 10 1
100 000 10 000

254

200 50 4

254 = 2 × 100 + 5 × 10 + 4 × 1
254 = 2 × 102 + 5 × 101 + 4 × 100

26 25 24 23 22 21 20

64 32 16 8 4 2 1

56 55 54 53 52 51 50

15625 3125 625 125 25 5 1

1324

125 75 10 4

2) State the place value of each digit in the numbers below.

a. 101112 b. 120213

c. 42317 d. 62008
62008 = 6 × 83 + 2 × 82 + 0 × 81 + 0 × 80
62008 = 6 × 512 + 2 × 64 + 0 × 8 + 0 × 1

62008 = 3072 + 128 + 0 + 0
62008 = 3200

10102 = 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20
10102 = 1 × 8 + 0 × 4 + 1 × 2 + 0 × 1
10102 = 8 + 0 + 2 + 0
10102 = 10

3) State the value of underlined digit in the numbers below.

a. 4015 b. 231014
4 × 52 3 × 43

= 4 × 25 = 3 × 64

= 100 = 192

c. 30157 d. 110012

e. 67108 f. 72489

g. 25876
5 × 103

= 5 × 1000

= 5000

4) Determine the values of the following numbers in base ten.
a. 100112

b. 2156

c. 23104

d. 21048

5) Convert 12310 of the following numbers base ten to a number in the base stated.

a. base five

method 1: (division using place value) Method 2: (division using base value)

Place 125 25 5 1
value 3/1
step 123/25 23/5

Base 5 443
Ans 4435

b. base eight

Place 512 64 8 1
value 123/64 59/8 3/1

step

Base 8 173
Ans 1738

c. base seven

Place 343 49 7 1
value 123/49 25/7 4/1

step

Base 7 234
Ans 2347

123 - base 7

7 123
7 17 …4
7 2 …3

0 …2

6) Convert 52010 of the following numbers base ten to a number in the base stated.

a. base two b) base six c) base four

Converting a number in certain base to base ten and then to another base

Base p Base 10 Base q

7) Convert 235 to certain base.
a. base 2

b. base 5
c) base 8

Converting a number in base two to base eight.
9) Convert the numbers in base two to numbers in base eight.
a)1100112

b) 1011012

c. 101101012

Converting a number in base eight to base two
10) Convert the numbers in base eight to numbers in base two.
a. 1458

b. 6508

c. 7018

Adding and subtraction of numbers in various base.

11) Calculate each of the following.

a) 1112 + 1102

111

+1 11 1 0

1101

b) 3758 + 678 375
+ 1 61 7

464

c) 20345 -1425

2 1 0+5 4 3+5 4

- 1 42

13 42

d) 15407- 2517

e) 100112-1102

12) Problem solving.
a. The price of a car is RM207650. If the price is in base eight, how much is the real price of the car?

a. Table below shows the favourite foods of 200 students in college Vista.

Food Percentage (%)

Pizza 25

Burger 40

Fried mee 17

Pasta 18

i) State the number of students who likes pizza and burger in base four.

ii) Calculate the total number of students who likes pizza and burger in

base four.

iii) Calculate the difference in number of students, in base seven,

between those who likes fried mee and pasta.