Number vs. Digit

What are digits?
Digits are the different unique values of a number system.

What are numbers?
Numbers are combinations of digits.

What is a number system?
A number system is a set of rules for forming numbers using a set of digits.

How many unique digits are in the binary number system?
There are 2 unique digits in the binary number system.

How many unique digits are in the octal number system?
There are 8 unique digits in the octal number system.

How many unique digits are in the decimal number system?
There are 10 unique digits in the decimal number system.

How many unique digits are in the hexadecimal number system?
There are 16 unique digits in the hexadecimal number system.

Forming a number from Digits

What is positional encoding?
Positional encoding is the method of forming a number from digits based on the relative position of each digit.

What value does each number system have?
Each number system has a base value.

Typically, what does the base value of a number system match?
The base value of a number system matches the number of unique digits.

What is each digit in a number multiplied by?
Each digit in a number is multiplied by integer exponents of the base depending on its relative position.

What is the rightmost digit multiplied by?
The rightmost digit is multiplied by .

What is the second rightmost digit multiplied by?
The second rightmost digit is multiplied by .

After the digits of a number are multiplied by the integer exponent of the base depending on their relative positions, how is the value of the number determined?
After the digits of a number are multiplied by the integer exponent of the base depending on their relative positions, the value of the number is determined by adding them together.

What does 11 represent in different number systems?

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What is a bit?
A bit is a binary digit.

How many bits are represented by one hexadecimal digit?
One hexadecimal digit represents four bits.

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Bit positions

Which bit is the Least Significant Bit (LSB)?
The LSB is the rightmost bit.

Which bit is the Most Significant Bit (MSB)?
The MSB is the leftmost bit.

Signed Number

What is the sign of a binary number if the sign digit is 0?
The sign of a number if the sign digit is 0 is positive.

What is the sign of a binary number if the sign digit is 1?
The sign of a number if the sign digit is 1 is negative.

What is the position of the sign digit in a binary number?
The position of the sign digit in a binary number is left of the MSB.

What digits represent the number in a signed binary number of length?
The digits that represent the number in a signed binary number is bits from the right.

Unsigned Addition

In what two ways are binary addition and decimal addition similar?
Binary addition is similar to decimal addition in that:

  1. Both are performed digit-wise.
  2. Digits are carried from the right to the left.

What happens to the carry digit in binary addition?
In binary addition, the carry digit is propagated from the LSB to the MSB.

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Half Adder

What does a half adder take as input?
A half adder takes two inputs, and .

What is the truth table for the half adder?
The truth table for the half adder is:

Sum () Carry
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1

What is the circuit diagram for the half adder?
The circuit diagram for the half adder is:
1-Bit Half Adder.png

Full Addition

What does a full adder take as input?
A full adder takes three digits, , , and as input.

What is the difference between the half adder and the full adder?
The difference between the half adder and the full adder is that the full adder takes the carry out from another adder while the half adder doesn't.

What does a full adder output?
A full adder outputs a sum and a carry out .

How can you create a multi-bit adder?
You can create a multi-bit adder by cascading full adders.

Multi-bit Adder

How many full adders do you need to chain together in order to add a pair of -bit binary numbers?
To add a pair of -bit binary numbers, you need to chain together full adders.

What is a ripple carry adder?
A ripple carry adder is a logic circuit that adds -bit binary numbers using multiple full adders.

Which adder do you need to use for the Least Significant Bit (LSB) of a binary number?
The adder you need to use for the LSB of a binary number is a half adder.

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K-mapping a Full Adder

Todo

This section has a Karnaugh map for a full adder's sum output and the carry out .

What is the Boolean equation for the sum of a full adder?
The Boolean equation for the sum () of a full adder is .

What is the Boolean equation for the carry out of a full adder?
The Boolean equation for the carry out () of a full adder is .

Full Adder Circuit

What is the circuit diagram for the full adder?
The circuit diagram for the full adder is:
1-Bit Full Adder.png