Question

all 4 - bit numbers: fill in the binary equivalents for the decimal numbers below. weve started the first three for you.
binary: 4 - bit number | decimal
0000 | 0
0001 | 1
0010 | 2
what do you notice when you compare the odd numbers with the even numbers? what might explain this?
binary numbers with exactly one 1: complete the chart with all 8 - bit binary numbers that have exactly one 1. weve done the first two for you.
binary: 8 - bit number (with exactly one 1) | decimal
0000 0001 | 1
0000 0010 | 2
0000 0100
what do you notice about the decimal equivalents above?

Explanation:

Step1: Recall binary - decimal conversion rule

The decimal value of a binary number $b_{n}2^{n}+b_{n - 1}2^{n - 1}+\cdots +b_{1}2^{1}+b_{0}2^{0}$, where $b_{i}\in\{0,1\}$.

Step2: Complete 4 - bit binary - decimal conversion

For 4 - bit binary numbers:

• $0011$: $0\times2^{3}+0\times2^{2}+1\times2^{1}+1\times2^{0}=0 + 0+2 + 1=3$
• $0100$: $0\times2^{3}+1\times2^{2}+0\times2^{1}+0\times2^{0}=0 + 4+0 + 0 = 4$
• $0101$: $0\times2^{3}+1\times2^{2}+0\times2^{1}+1\times2^{0}=0 + 4+0 + 1=5$
• $0110$: $0\times2^{3}+1\times2^{2}+1\times2^{1}+0\times2^{0}=0 + 4+2 + 0 = 6$
• $0111$: $0\times2^{3}+1\times2^{2}+1\times2^{1}+1\times2^{0}=0 + 4+2 + 1=7$
• $1000$: $1\times2^{3}+0\times2^{2}+0\times2^{1}+0\times2^{0}=8 + 0+0 + 0 = 8$
• $1001$: $1\times2^{3}+0\times2^{2}+0\times2^{1}+1\times2^{0}=8 + 0+0 + 1=9$
• $1010$: $1\times2^{3}+0\times2^{2}+1\times2^{1}+0\times2^{0}=8 + 0+2 + 0 = 10$
• $1011$: $1\times2^{3}+0\times2^{2}+1\times2^{1}+1\times2^{0}=8 + 0+2 + 1=11$
• $1100$: $1\times2^{3}+1\times2^{2}+0\times2^{1}+0\times2^{0}=8 + 4+0 + 0 = 12$
• $1101$: $1\times2^{3}+1\times2^{2}+0\times2^{1}+1\times2^{0}=8 + 4+0 + 1=13$
• $1110$: $1\times2^{3}+1\times2^{2}+1\times2^{1}+0\times2^{0}=8 + 4+2 + 0 = 14$
• $1111$: $1\times2^{3}+1\times2^{2}+1\times2^{1}+1\times2^{0}=8 + 4+2 + 1=15$

Step3: Complete 8 - bit binary - decimal conversion with exactly one 1

• $00001000$: $0\times2^{7}+0\times2^{6}+0\times2^{5}+0\times2^{4}+1\times2^{3}+0\times2^{2}+0\times2^{1}+0\times2^{0}=8$
• $00010000$: $0\times2^{7}+0\times2^{6}+0\times2^{5}+1\times2^{4}+0\times2^{3}+0\times2^{2}+0\times2^{1}+0\times2^{0}=16$
• $00100000$: $0\times2^{7}+0\times2^{6}+1\times2^{5}+0\times2^{4}+0\times2^{3}+0\times2^{2}+0\times2^{1}+0\times2^{0}=32$
• $01000000$: $0\times2^{7}+1\times2^{6}+0\times2^{5}+0\times2^{4}+0\times2^{3}+0\times2^{2}+0\times2^{1}+0\times2^{0}=64$
• $10000000$: $1\times2^{7}+0\times2^{6}+0\times2^{5}+0\times2^{4}+0\times2^{3}+0\times2^{2}+0\times2^{1}+0\times2^{0}=128$

Step4: Analyze odd - even in binary

In binary, an even number ends with 0 and an odd number ends with 1. This is because an even number $n$ can be written as $n = 2k$ for some integer $k$. In binary, multiplying by 2 is equivalent to shifting the binary digits to the left. So, an even number in binary will have 0 in the $2^{0}$ place. An odd number $m=2k + 1$, so it will have 1 in the $2^{0}$ place.

Step5: Analyze 8 - bit binary with one 1

The decimal equivalents of 8 - bit binary numbers with exactly one 1 are powers of 2. This is because when there is only one 1 in the binary number, say in the $2^{n}$ place, the decimal value is $2^{n}$.

Answer:

4 - bit binary - decimal pairs:

Binary: 4 - bit number	Decimal
0100	4
0101	5
0110	6
0111	7
1000	8
1001	9
1010	10
1011	11
1100	12
1101	13
1110	14
1111	15

8 - bit binary - decimal pairs with exactly one 1:

Binary: 8 - bit number (with exactly one 1)	Decimal
00010000	16
00100000	32
01000000	64
10000000	128

Observation about odd - even: Even binary numbers end with 0, odd binary numbers end with 1.
Observation about 8 - bit binary with one 1: Decimal equivalents are powers of 2.