Mathematics

                                                                                               12/10/2020

TOPIC: NUMERATION SYSTEM (NUMBER BASES)

SUB-TOPIC: Converting from base ten to other bases 

Introduction

Numbers are usually written in base 10(decimal numeral) but they can be written or converted to other bases as well. Base ten has only 10 digits i.e. 0,1,2,3,4,5,6,7,8,9. To indicate that a base other than base ten is being used, a small subscript is added after the number. Hence 425 or 42𝑓𝑖𝑣𝑒 indicates that the number has been written in base 5 and it is read as “four two base five”.

Numerals of number bases

Base two/binary system: This has only two digits i.e. 0 and 1. The first ten whole numbers in base two are 0,1,10,11,100,101,110,111,1000,1001.

Base three system: The numeration system in base three has only 3 digits i.e. 0,1 and 2. The first ten whole numbers in base three are 0,1,2,10,11,12,20,21,22,100.

Base five system: This has only five digits i.e. 0,1,2,3 and 4. The first ten whole numbers in base five are 0,1,2,3,4,10,11,12,13,14.

Base six system: The numeration system in base six has only 6 digits i.e. 0,1,2,3,4, and 5. The first ten whole numbers in base six are 0,1,2,3,4,5,10,11,12,13.

Converting from base 10 to other bases

In order to change from base 10 to a different base, a method involving successive division is used. The given decimal numeral is divided repeatedly by an appropriate base number and the remainders including zero are noted at each stage. The division is continued until nothing is left to divide. The answer is obtained by reading the remainders upward as indicated by the arrows in the examples that follows. 

Example 1

Convert 84𝑡𝑒𝑛 to base 5.

Solution

5 84 Rem

5 16 4

5 3 1

0 3

84𝑡𝑒𝑛= 314𝑓𝑖𝑣𝑒

Example 2

Convert 25𝑡𝑒𝑛 to base 2.

Solution

2 25 Rem

2 12 1

2 6 0

2 3 0

2 1 1

0 1

25𝑡𝑒𝑛 = 11001𝑡𝑤𝑜

Example 3

Convert 226𝑡𝑒𝑛 to base 8.

Solution

8 226 Rem

8 28 2

8 3 4

0 3

226𝑡𝑒𝑛 = 342𝑒𝑖𝑔ℎ𝑡

Try 

Convert 180𝑡𝑒𝑛 to base 5.

Solution

5 180 Rem

5 36 0

5 7 1

5 1 2

0 1

180𝑡𝑒𝑛= 1210𝑓𝑖𝑣𝑒

Try 

Convert 123𝑡𝑒𝑛 to base 2.

Solution

2 123 Rem

2 61 1

2 30 1

2 15 0

2 7 1

2 3 1

2 1 1

0 1

123𝑡𝑒𝑛 = 1111011𝑡𝑤𝑜

15/10/20

TOPIC:NUMERATION SYSTEM (NUMBER BASES)

SUB-TOPIC: Converting from other bases to base ten (decimal Numeral)

Convert 110112 to a decimal numeral.

Numbering   4    3    2   1    0

Digits 1  1  0  1  1

Use this listing to convert each digit to the power of two that it represents:

4   3  2  1   0

110112 =(1x24)+(1x23)+(0x22)+(1x21)+(1x20)

=(1x16)+(1x8)+(0x4)+(1x2)+(1x1)

=16+8+0+2+1

= 27

Convert 12425 to a base ten numeral. 

Solution 

3 2 1 0

12425 =(1x53)+(2x52)+(4x 51)+(2x 50)

=(1x125)+(2x25)+(4x5)+(2x 1)

=125+ 50 + 20 + 2

=197

Convert 4536 to base ten.

Solution

2 1  0

4536 = (4x62)+(5x61)+(3x60)

=(4x36)+(5x6)+(3x1)

=144+30+3

= 177

 

Convert 1101012 to a decimal numeral.

Solution

5  4  3   2  1  0

1101012 =(1x25)+(1x24)+(0x23)+(1x22)+(0x21)+(1x20)

=(1x32)+(1x16)+(0x8)+(1x4)+(0x2)+(1x1)

=32+16+0+4+0+1

= 53

Convert 2302𝑓𝑜𝑢𝑟 to a decimal numeral.

 

3    2    1    0

23024

Solution

= (2x43)+(3x42)+(0x41)+(2x40)

=(2x64)+(3x16)+(0x4)+(2x1)

= 128+ 48 + 0 + 2

= 178

 

Convert 765𝑒𝑖𝑔ℎ𝑡 to a decimal numeral.

Solution

2    1    0

7658 = (7x82) +(6x81)+(5x80)

= (7x 64)+ (6x8) +(5x1)

= 448+ 48 + 5

= 501

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