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The decimal number system has 10 symbols, octal has 8,
binary has 2, hexadecimal 16. One can easily create a
number system using any number greater than 1 as its base.
For example,
Base 17 {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,G}
base 16 {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}
base 12 {0,1,2,3,4,5,6,7,8,9,A,B}
base 10 {0,1,2,3,4,5,6,7,8,9}
base 8 {0,1,2,3,4,5,6,7}
base 3 {0,1,2}
base 2 {0,1}
Why is a number system’s base not a symbol in the number
system’s set of symbols?
If the base were also a symbol then the representation of a
number as the sum of the products of the symbols and powers
of the base would become more complicated. A symbol’s value
would no longer be a function of its position in the
number. Because the base is not a symbol, a symbol’s value
is very simple: it is the symbol followed by zeroes. How
many zeroes depend on the symbol’s position in the number.
The good thing about representing numbers this way is the
symbol in one position cannot affect the symbols in the
other positions. For example,
378 = 300 + 70 + 8
in some position based number system. Note that replacing
the 3 with a 6 does not affect the other symbols. The new
number is
678 = 600 + 70 + 8
We perform these expansions by multiplying each symbol by a
power of 10 where the value of “10” is ten only in the
decimal number system. If we are working with a number
system, other than base 10 number system then the value of
“10” is this number system is b in the base 10 number
system.
Suppose, for example, that the value of b is “12”.
37812 =
=
=
=
3 x 102 + 7 x 101 + 8 x 100 in base 12
3 x 122 + 7 x 121 + 8 x 120 in base 10
3 x 144 + 7 x 12 + 8 x 1 in base 10
52410
A related question is why do symbols in different number
systems have the same value. For example, why is 310 = 312?
They have the same value because it is convenient for us
that they have the same value. We could create completely
distinct sets of symbols for each number system but then we
would have to define how to translate one set of symbols
into another set of symbols. We do it when we convert roman
numbers into decimal numbers or decimal numbers to roman
numbers. But it is extra work.
How do we convert from one base to another base? We convert
the number from the first base to base 10 and then convert
the number from base 10 to the second base
98712 = ?8
98712 = 9 x 12 x 12 + 8 x 12 + 7 x 1
= 9 x 144 + 8 x 12 + 7 x 1
= 1296 + 96 + 7
= 139910
An algorithm for converting from base 10 to base n
divides the number repeatedly by the base until the
quotient is zero. The converted number is the remainders of
the division operation.
1399 = 8 x 174 + 7
174 = 8 x 21 + 6
21 = 8 x 2 + 5
2 = 8 x 0 + 2
98712 = 25678
Why does this algorithm work?
1399 = 8 x 174 + 7
= 8 x (8 x 21 + 6) + 7
= 8 x (8 x (8 x 2 + 5) + 6) + 7
= 8 x 8 x 8 x 2 + 8 x 8 x 5 + 8 x 6 + 7
= 2 x 83 + 5 x 82 + 6 x 81 + 7 x 80 in base 10
= 2 x 103 + 5 x 102 + 6 x 101 + 7 x 100 in base 8
= 25678
convert 183 in base 10 to base 12
183 = 12 x 15 + 3
15 = 12 x 1 + 3
1 = 12 x 0 + 1
18310 = 13312
183 =
=
=
=
=
=
12 x 15 + 3
12 x (12 x 1 + 3) + 3
12 x 12 x 1 + 12 x 3 + 3
1 x 122 + 3 x 121 + 3 x 120
13312
An algorithm for converting from base n to base 10 expands
the number as the sum of products of the symbols and powers
of the base and then replaces the base with its value in
base 10.
e.g. convert 13312 to base 10
13312 = 1 x 102 + 3 x 101 + 3 x 100
= 1 x 122 + 3 x 121 + 3 x 120
= 144 + 36 + 3
= 183
convert 17710 to base 8
177 = 22 x 8 + 1
22 = 2 x 8 + 6
2 = 0 x 8 + 2
17710 = 2618
convert 1778 to base 10
1778 = 1 x 102 + 7 x 101 + 7 x 100
= 1 x 82 + 7 x 81 + 7 x 80
= 1 x 64 + 7 x 8 + 7 x 1
= 64 + 56 + 7
= 12110
Algebra of Exponents
Why is a0 = 1 for all a  0?
am x an = an+m?
a2 x a3 = a x a x a x a x a = a5
That is, a2 x a3 = a2+3 = a5
By convention, a-n = 1/an
Moreover, am/an = am-n
Suppose m = n then am/an = am-n = a0 = 1
Fractions
A fraction is the ratio of two integers where the integer
in the fraction’s denominator is not zero. A fraction has a
representation has a ratio and as a decimal fraction.
Terminating Factions
1/5 = 0.20
Continuous Fractions
1/3 = 0.333...
Rules and Conventions
1.000... is not a continuous decimal fraction. Why?
0.999... is not a continuous decimal fraction. Why?
How do you convert a fraction from a ratio representation
to a decimal fraction representation?
How do you convert a fraction from a decimal fraction
representation to a ration representation?
Why are some fractions continuous decimal fractions?
Why are some fractions terminating decimal fractions?
How do you know that a fraction has a continuous decimal
fraction representation?
How do you know that a fraction has a terminating decimal
fraction representation?
Is it possible that a fraction has a terminating decimal
fraction representation in one number system and a
continuous decimal fraction representation in another
number system?
Class Exercises
Convert 1002 to base 3
Convert 1002 to base 4
Convert 1002 to base 8
Convert 1002 to base 10
Convert 1002 to base 12
Convert 1003 to base 2
Convert 1004 to base 2
Convert 1008 to base 2
Convert 10012 to base 2
Can we convert 1288 to base 10?
Can we convert 2012 to base 10?
Does 1/7 have a terminating decimal fraction
representation?
Does 1/5 have a termination decimal fraction
representation?
Convert 0.333... from base 10 to base 2
Convert 0.333... from base 10 to base 3
modular arithmetic
modular arithmetic always has two operations – addition,
and multiplication; always has subtraction but usually
ignores subtraction; and sometimes permits division under
certain circumstances.
Modular arithmetic is useful in number systems with a
finite number of numbers.
Mod
0 x
0 x
0 x
1 x
1 x
1 x
2 x
2 x
2 x
3
0
1
2
0
1
2
0
1
2
=
=
=
=
=
=
=
=
=
=
{0, 1, 2}
0
0
0
0
1
2
0
2
1
Why does 2 x 2 = 1?
Modular arithmetic is like ordinary arithmetic with an
extra step: we perform the operation, get the result,
divide the result by the modulus, in this case “3”, and the
remainder is the final result.
For example, 2 x 2 = 4, the intermediate result. When we
divide 4 by 3 the remainder is 1, our final result.
Z3 = {0, 1, 2}
0 + 0 = 0
0 + 1 = 1
0 + 2 = 2
1 + 0 = 1
1 + 1 = 2
1 + 2 = 0
2 + 0 = 2
2 + 1 = 0
2 + 2 = 1
How do we define subtraction?
In any number system, for every a in the number system
there is a b such that a + b = 0. We then say that a = -b.
From the addition table we see that
0 = -0
1 = -2
2 = -1
We also
0 = 0 –
0 = 1 –
0 = 2 –
1 = 1 1 = 2 1 = 0 2 = 2 2 = 0 2 = 1 -
define subtraction from the addition table:
0
1
2
0
1
2
0
1
2
An integer’s divisions are those numbers that divide it
without remainder.
E.g.
12’s divisors are 2, 3, 4, and 6.
8 and 9 are not divisors of 12.
A prime number is any positive integer that has exactly 2
distinct divisors - 1 and itself. Is 1 a prime number?
Do we get different kinds of results if the modulus is
prime or not prime?
Some modulus number system can define the operation of
division. A modulus number system can divide if a x b = 0
means that a or b = 0.
This is not true when the modulus is 4 because 2 x 2 = 0 in
the modulus system. It is true whenever the modulus is a
prime number.
Suppose
1 x 1 =
1 x 2 =
2 x 1 =
2 x 2 =
the modulus is 3.
1
2
2
1
We can define division
1 / 1 = 1
2 / 2 = 1
2 / 1 = 2
1 / 2 = 2
Why? If a x b = c then a = c / b
Define subtraction and division in modulo 7
Define subtraction and division in modulo 8
What criteria to you use to determine that 11 divides a
number?
What criteria do you use to determine that 4 divides a
number?