We represent our base 10 numbers, such as 201 in the following way: [1.1]
So
that each digit is in 0..9, and its position determines which power of
10 we need to multiply it by. Two important symbols in our decimal
system are 0, which has been used extensively only in the last few
hundred years in Europe, and ".", the decimal point which marks the
whole numbers from the fractions (something which was absent in the
Babylonian system, and which causes some confusion.. when, for
instance, we cannot distinguish 0.25 and 25!)
In general, we represent the whole numbers in our base 10 system in the following way: [1.2] Where a is a number, such as 201, and the a's with subscripts are numbers between 0 and 9.
Generalising even further, we can represent numbers in the base, b, as follows: [1.3]
If we divide a number a, by b continuously, we find that the remainders of division are the digits of our new number base b.
When
converting from one base to another (excluding base 10) we need to do
arithmetic in a base other than ten. For this reason it is better to
convert from one base to another through the intermediary of base 10;
that is, we convert one number in base b1 into base 10 and then from base 10, we convert the number to b2.
Converting Whole Numbers in Base 10 to Another Base
We can convert a number in base 10 to another base system by repeatedly dividing the number by b, the new base. For instance, we wish to convert 10 in base 10, to a number in base 3.
Convert 10, base 10 to base 3
Number of 3's
Remainder
10=
3•3+
1
3=
3•1+
0
1=
3•0+
1
We read off the new number, base 3 from the remainders: [1,0,1], or 1013. So 10 in base 10 is 101 in base 3.
Remark,
we could have noticed that because the 1 in the last but one line is
less than 3, we could have read the number off beginning with this
obvious next remainder, and the other remainders: 101, base 3.
As a second example, we can convert 102 in Base 10 to Base 16 (or hexadecimal).
Convert 102, base 10 to base 16
Number of 3's
Remainder
102=
16•6+
4
6=
16•0+
6
Reading off the remainders, we note that 10210=6416
Converting a Number in Base b to a Decimal Number
A number, such as 101 in base 2 can be thought of as 12011,
where the superscripts are reminders that in the position to the right
of the index, the number in base 10 is that number times that power of
two. That is: 12011=1•4+0•2+1=6.
And for 64 in base 16: 6140=6•16+4•160=100, base 10 And 101, base 3 gives: 120110=1•32+0•31+1•30=10, base 10
Converting a Fraction in Base 10 to Base b
A decimal fraction such as 0.102, can be written in terms of negative powers of 10: [2.1]
In general, a fractional number, a, in base b can be written: [2.1] In decreasing powers of ten, with the first position being 10-1.
Or, alternatively written as: [3.2] In
converting a decimal in base 10 to a fraction in base b, we do the
opposite of converting the whole numbers: we multiply the number
in base 10 by b repeatedly, and our remainder is the integer part of the resulting number.
For instance, 0.5 in base 10 is 4•0.5=2, or 0.5 in base 4 is 0.2 in base 4
Convert 0.5 Base 10 to Base 4
Integer, Number of 4's
Fraction
0.5•4
2
0
0.5 in base 10 is 0.2 in base 4 )
Convert 0.5 Base 10 to Base 3
Integer Number of 3's
Fraction
0.5•3=
1
.5
0.5•3=
1
.5
0.510 in base 3 is therefore 0.111...3.
Convert 0.16 Base 10 to Base 8
Integer, Number of 8's
Fraction
0.16•8=
1
.28
0.28•8=
2
.24
0.24•8=
1
.92
0.92•8=
7
.36
0.36•8=
2
.88
0.88•8=
7
.64
0.16 base 10 is 0.121727... base 8
Converting a Fraction Base b to Base 10
To convert a fraction base b, to base 10, we recall: [3.2, repeated]
For instance, 0.2 in base 4 is 2/4 in base 10, or 0.5.
Another example: 0.121727 in base 8 is
(to base 10) is, by spreadsheet:
Ken's book is packed with examples and explanations that enable you to discover more than 150 techniques to speed up your arithmetic and increase your understanding of numbers. Paperback and Kindle: