ECE 275: Place value number system
Announcments
- Please wear masks properly.
- Midterm is on Oct 6th, 9 AM in class.
Functional decomposition Ex4
| \(\bA \) | |||||
|---|---|---|---|---|---|
| \(\bB\) | \(B \) | ||||
| \(\bC\) | \(C \) | \(\bC\) | |||
| \(\bD\) | \(\bE\) | 0 | 1 | 1 | 1 |
| \(E\) | 1 | 0 | 0 | 0 | |
| \(D\) | 0 | 1 | 1 | 1 | |
| \(\bE\) | 1 | 0 | 0 | 0 | |
| \(\bA \) | |||||
|---|---|---|---|---|---|
| \(\bB\) | \(B \) | ||||
| \(\bC\) | \(C \) | \(\bC\) | |||
| \(\bD\) | \(\bE\) | 1 | 0 | 0 | 0 |
| \(E\) | 0 | 1 | 0 | 0 | |
| \(D\) | 1 | 0 | 0 | 0 | |
| \(\bE\) | 0 | 1 | 0 | 0 | |
Press b to see blackboard
Place value number system
- Decimal: Base 10
- Binary: Base 2
| Decimal | Binary |
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
| 8 | 1000 |
| 9 | 1001 |
| 10 | 1010 |
Binary to decimal
Decimal to binary
Value of a number
- A number system with base \( b \) has at least \( b \) unique symbols with special.
- A number system is has two operators \( + \) and \( \cdot \). The number system must be closed under the two operators.
- There is at least one symbol \( 0 \) such that \( x + 0 = x \)
- There is at least one symbol \( 1 \) such that \( x \cdot 1 = x \)
- A number in base \( b \) written as \( d_n \dots d_1 d_0 \) has the value \[ d_n \dots d_1 d_0 = d_n b^n + \dots + d_1 b^1 + d_0 b^0 \]
Hexadecimal
| Decimal | Hexadecimal |
| 0 | 0 |
| \(\vdots \) | \(\vdots\) |
| 9 | 9 |
| 10 | A |
| 11 | B |
| 12 | C |
| 13 | D |
| 14 | E |
| 15 | F |
Hexadecimal to binary
Arbitrarily new number system
Define two symbols \( 0, 1 \) as required.
- ♥ = 0
- ♦ = 1
Define rest of the symbols in increments of \( ♦ \)
- ♣ = ♦ + 1
- ♠ = ♣ + 1
The value of ♠♦♥ = \( 3\times 4^2 + 1 \times 4^1 + 0 \times 4^0\)
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