1.1 number bases

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Numbers in Base Two

Number

Place value of numbers in base 2

2^4=16

2^3=8

2^2=4

2^1=2

2^0=1

0 01 12 1 03 1 14 1 0 05 1 0 16 1 1 07 1 1 18 1 0 0 09 1 0 0 110 1 0 1 0


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Numbers in base ei ht

Number

Place value of numbers

in base 88^4=0 6

8^3=12

8^2=6

8^1=8

8^0=1

0 01 1

2 2

456 678 1 0

1 110 1 228 1 2


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Numbers in base Five

Number

Place value of numbers

in base 55^4=62

5^3=12

5^2=2

5^1= 5^0=1

0 01 1

2 23 34 4

5 1 06 1 17 1 28 1 39 1 4

10 2 0

2 1 0 28


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Value of a digit of a

number in base 2,8 and 5Example:

1. 11012 = 1 x 22 = 4

2. 50368 = 5 x 83 =2560

3. 24105= 4 x 52 = 100


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Writing Numbers in Base 2, 8 and 5 i

n

Expanded Notation

Examples:

1. 10112= 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20

2. 3478 = 3 x 82 + 4 x 81 + 7 x 80

3. 1425= 1 x 52 + 4 x 51 + 2 x 50


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1.1d Converting Numbers in

Base Two, Eight and Five toBase 10 and vice Versa


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Steps to Remember:

1. Write the number in expanded notation

2. Simplify the expanded notation into a single number.

Examples:

(i) 10112 = 1 x 23 + 0 x 22 + 1 x21 +1 x 20 =1110

(ii) 2768 = 2 x 82 + 7 x 81 + 6 x 80=19010

(iii) 1345 = 1 x 52 + 3 x 51 + 4 x 50 =4410


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Steps to remember to convert a number in base 10 tonumber in base 2, 8 or 5

(i) Perform repeated division until the quotient is 0.

(ii) Write the number in the new base by referring tothe remainders from bottom to the top.

Example:

(a) Convert 910 to base 22 92 4R 1

2 2R 02 1R 0

0R 1 910 = 10012


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(a) Convert 2810 to base 88 288 3 R4

8 0R 3

2810 = 348

(b) Convert 7610 to base 5

5 765 15R15 3R0

0R3 7610 = 3015


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1112 = 125


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435 = 101112


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1378= 3405


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1378= 3405


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100 111 000 011


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11

1+1 = 210 = 102


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2020


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1.1 number bases

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