Positional number systems
Any number written in the positional number system
p can be represented in expanded form. The expanded form of the number – this
record in form bit terms , written using the degree of the corresponding category and the base of the degree (the base of the account).
\(A_p = \overline{a_4a_3a_2a_1a_0 }= a_4 \cdot p^4 + a_3 \cdot p^3 + a_2 \cdot p^2 + a_1 \cdot p^1 + a_0 \ cdot p^0\)
\(p^0 = 1\)
How to calculate the digits of a number in decimal notation
\(a_0 = A\ mod\ 10\), where
mod is the remainder of A divided by 10.
\(\overline{a_1a_0} = A\ mod\ 100\)
etc.
(in other number systems, we will divide by the base of the number system to the appropriate degree - knowledge of this will be useful in a software solution)
Properties
| |
For decimal number system (p = 10) |
For binary number system (p = 2) |
| 1 |
\(10 ^ N = 1\underbrace{0...0}_{N}\) |
\(2 ^ N = 1\underbrace{{0...0}_2}_{N}\) |
| 2 |
\(10 ^ N-1 = \underbrace{9...9}_{N}\) |
\(2 ^ N-1 = \underbrace{{1...1}_2}_{N}\) |
| 3 |
\(10 ^ N-10 ^ K = 10^K \cdot (10^{N-K}-1) = \underbrace{9...9}_{N-K }\underbrace{0...0}_{K} \) |
\(2 ^ N-2 ^ K = 2^K \cdot (2^{N-K}-1) = \underbrace{1...1}_{N-K }\underbrace{0...0}_{K} \)(\(K<N\)) |
| 4 |
|
\(2 ^ N+2 ^ N = 2^{N+1}\),
therefore
\(2 ^ N= 2^{N+1}-2^N\)
\(-2 ^ N= -2^{N+1}+2^N\) |
In general
| |
For a number system with base p |
| 1 |
\(p ^ N = 1\underbrace{{0...0}_p}_{N}\) |
| 2 |
\(p ^ N-1 = \underbrace{{(p-1)...(p-1)}_2}_{N}\) |
| 3 |
\(p ^ N-p ^ K = p^K \cdot (p^{N-K}-1) = \underbrace{(p-1)...(p- 1)}_{N-K}\underbrace{0...0}_{K} \) |
| 4 |
|
Analytical solution algorithm
- Represent all numbers in an expression as powers of the specified number system.
- Regroup the terms in the expression so that the powers are in descending order.
- For a binary number system, using property 4, rewrite the expression so that the sign "
-" alternated with "+".
- Using properties 1, 2, 3, find the answer.