Write the following numbers in the requested bases.
a. 10=_________base 2
b. 42=_________base 2
c. 10101011 base 2=________base 10
d. 77=_______base 5
e. 40 base 6=_______base 10
base 2
1 = 2^0 which is 1 decimal (anything^0=1)
1 0 = 2^1 which is 2 decimal
1 0 0 = 2^2 which is 4 decimal
1 0 0 0 = 2^3 which is 8 decimal
1 + n zeros = 2^n
so
a. 10 = 8 + 2 decimal
8 is 2^3 1 0 0 0
2 is 2^1 0 0 1 0
so the answer is
1 0 1 0
b. 42 = 32 + 10 (we already know 10)
32 = 2^5 = 1 0 0 0 0 0
10 = __________1 0 1 0
so the answer is
1 0 1 0 1 0
c.
2^7 +0 + 2^5 +0 +2^4 +0 +0 +2^1 +2^0
128+32+16+2+1 = 179
d. 5^0=1
1 = 5^0
10 = 5^1
100 = 5^2 = 25
1000 = 5^3 = 125 ...
1 + n zeros = 5^n
well 77 = 3*25 + 2
= 3*5^2 + 2 * 5^0
3 0 0 + 2
3 0 2
e. 1 = 6^0
1 0 = 6^2 = 36
1 0 0 = 6^3 = 216 ....
so 4 0 is 4 * 36
which is 144
e.
1 = 6^0
1 0 = 6^1 = 6
1 0 0 = 6^2 = 36
1 0 0 0 = 6^3 =216
so 40 is 4*6 or 24
ARRGHHHH!!!!
Unless you are studying Computer Science and are working in base 2, 8 or 16, tell your teacher to quit teaching this silly stuff and get back to real Mathematics.
Tell him/her that an old mathteacher with over 33 years experience said so.
(sorry about the political bias)
LOL - well I think it is fun !
Had an extra 0 when I copied
c. 10101011 base 2=________base 10
2^7+0+2^5+0+2^3+0+2^1+2^0
128+32+8+2+1
= 171
a. To convert 10 to base 2 (binary), you can use the method of successive divisions. Start by dividing 10 by 2 and write down the remainder. Then, divide the quotient obtained by 2 again and write down the remainder. Repeat this process until the quotient becomes 0.
10 ÷ 2 = 5, remainder 0
5 ÷ 2 = 2, remainder 1
2 ÷ 2 = 1, remainder 0
1 ÷ 2 = 0, remainder 1
Now, read the remainders from the last to the first to get the binary representation:
10 = 1010 base 2
b. To convert 42 to base 2, follow the same method of successive divisions:
42 ÷ 2 = 21, remainder 0
21 ÷ 2 = 10, remainder 1
10 ÷ 2 = 5, remainder 0
5 ÷ 2 = 2, remainder 1
2 ÷ 2 = 1, remainder 0
1 ÷ 2 = 0, remainder 1
Reading the remainders from the last to the first gives:
42 = 101010 base 2
c. To convert 10101011 in base 2 to base 10, each digit represents a power of 2. Start from the rightmost digit and multiply it by 2 raised to the power of its position, which starts from 0.
1 × 2^0 = 1
1 × 2^1 = 2
0 × 2^2 = 0
1 × 2^3 = 8
0 × 2^4 = 0
1 × 2^5 = 32
0 × 2^6 = 0
1 × 2^7 = 128
Add up all the values:
10101011 base 2 = 1 + 2 + 0 + 8 + 0 + 32 + 0 + 128 = 171 base 10
d. To convert 77 to base 5, use the method of successive divisions:
77 ÷ 5 = 15, remainder 2
15 ÷ 5 = 3, remainder 0
3 ÷ 5 = 0, remainder 3
Read the remainders from the last to the first:
77 = 302 base 5
e. To convert 40 in base 6 to base 10, multiply each digit by 6 raised to the power of its position, starting from 0:
4 × 6^0 = 4
0 × 6^1 = 0
Add up the values:
40 base 6 = 4 + 0 = 4 base 10