Conversion from Other Bases to Base (10) Ten
It is possible to use any number as a base in building a numeral system. The number of unit used in the system is always equal to the base.
Decimal/Denary Base 10-Digits Quinary
Decimal/DenaryBase 10-Digits | QuinaryBase 5-Digits | BinaryBase 2-Digits | OctalBase 8-Digits | HexadecimalBase 16-Digits |
0 | 0 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 1 |
2 | 2 | 2 | 2 | |
3 | 3 | 3 | 3 | |
4 | 4 | 4 | 4 | |
5 | 5 | 5 | ||
6 | 6 | 6 | ||
7 | 7 | 7 | ||
8 | 8 | |||
9 | 9 | |||
10 (A) These are | ||||
11 (B) the current | ||||
12 (C) Conventional | ||||
13 (D)notations | ||||
14 (E) | ||||
15 (F) |
In the hexadecimal base (base 16) system the letter A represents 10, B represent 11, C represents 12, D
for 13, E for 14, F for 15 since these numbers are regarded as single digits in the hexadecimal scale.
These are the current conventional notations.
In order to differentiate the numbers in different bases we use subscript notation. For example, 258
reads as two, five to base 8 meaning 2 x 81 + 5 x 80 that is the 2 eights and 5 ones. This equals 1610 +
510 = 2110.
Note that the above expression as twenty eight base eight, because this is the language in the denary
system) but can also read as “two one to base 3”.
The powers correspond to the place values of the number and they show the order of the digits. i.e.
12203 = 1 x 33 + 2 x 32 + 2 x 31 + 0 x 30 ——————————- Expanded form
= 27 + 18 + 6 + 0 = 5110
Conversion by repeated/Successive Multiplication
This is another method of converting other bases to denary or decimals system is by repeated or successive multiplication. This method involves the successive multiplication of the digits by the given base. This is done starting with the first digit to the last and not as in ordinary multiplication where we multiply from the last digit to the first.
- Multiplying the first digit by 7, i.e. 1 x 7 = 7
- Add this to the second digit, i.e. 7 + 2 = 9
- Multiply this sum by 7, i.e. 9 x 7 = 63
- Add the third digit to this product, i.e. 67 x 4 = 67
- Multiply this sum by 7, i.e. 67 x 7 = 469
- Add the last digit to this product (469 + 6 = 475. This last sum is the required number in 10.
Conversion of Decimal Fraction in Any Base to Base 10
The fractional part must be expressed in decimal form before conversion is possible, this is because it is important to represent the place value of the number before conversion can be done.
Example: convert 6.47 to denary number.
We call this reference table i.e. the table showing the digit position and place value of the given numbers. And it makes for easy translation of the numbers in the fractional part to base 10.
7^{0} | 7^{-1} | 7^{-2} | |
Number | 6 | 4 |
In expanded form 6.4_{7} = 6 x 7^{0} + 4 x 7^{-1}
= 6 x 1 + 4 x 1/7
= 6 + 4/7 = 6.57_{10} (2 places of decimal).
32.51_{6}
6^{1} | 6^{0} | 6^{-1} | 6^{-2} | |
Number | 3 | 2 | 5 | 1 |
32.51_{6} = 3 x 6^{1} + 2 x 6^{0} + 5 x 6^{-1} + 1 x 6^{-2}
= 3 x 6 + 2 x 1 + 5 x 1/6 + 5 x 1/6^{2}
= 18 + 2 + 5/6 + 1/36
= (20 + 30+1/36)_{10} = 20.86_{10} (2 places of decimal)
Conversion from Base 10 to Other Bases
This is gotten by successive or continued division of the number (in base 10) by the base and writing out the remainders at each stage, until the quotient is 0
Convert 106_{10} to binary base
To do this conversion, I need to divide repeatedly by 2, keeping track of the remainders as I go. Watch below
Where Rem means remainder
2 106 Rem Interpretation
2 53 + 0 – (53 x 21+0 x 20)
2 26 + 1 – (26 x 22+1 x 21)
2 13 + 0 – (13 x 23+0 x 22)
2 6 + 1 – (6 x 24 +1 x 23)
2 3 + 0 – (3 x 25+ 0 x 24)
2 1 + 1 – (1 x 26+ 1 x 25)
2 0 + 1 – (0 x 27+1 x 26)
At each level in the division both the quotient and the remainder are expressed in the expanded powers of two.
Hence 106_{10} = 1 x 2^{6} +1 x 2^{5} +1 x 2^{4} +1 x 2^{3} +1 x 2^{2} +1 x 2^{1} +1 x 2^{0}
This means expressing the remainders in powers of two. So we read the remainders upwards because we read numbers from the largest to the least and the last remainder is the largest number in that base.
Hence 106_{10} = 1101010_{2}
Conversion from One Base to Another
Here we shall discuss the conversion from one base say quinary (base 5) to octal.
To do this we convert the given number first to denary (base 10), then to the required base.
Example: Express 4140_{five} to Octal.
Solution: 4104_{five} to denary
4 x 5^{3} + 1 x 5^{2} + 0 x 5^{1} + 4 x 5^{0}
4 x 125 + 1 x 25 + 0 + 4
500 + 25 + 4 = 529_{10}
Then convert 529_{10} to Octal
8 529
8 66 + 1
8 8 + 2
8 1 + 0
8 0 + 1
52910 = 1021_{eight}
i.e. 4104_{five} = 1021_{eight}
EXERCISES
Lets see how much you’ve learnt, attach the following answers to the comment below
Write the following in expanded form by counting the position of the digits
- 2481
- 60367
- Express 63710 in the expanded form and find the value in base 10. Change the number to denary numbers (base 10)
- Convert 32.51_{6} to denary number.
- Express 143_{6} in base 7
1 thought on “Classwork Series and Exercises {Mathematics- SS1}: Conversion from Other Bases to Base (10) Ten”
Convert 17eight To Base Ten