Convert the following numbers as indicated in the:
i. 10101112 into decimal.
ii. 1111011.1112 into decimal.
iii. 435.12510 into binary.
iv. 584610 into hexadecimal system. (
i. To convert the number 1010111 to decimal, we start from the right and multiply each digit by the corresponding power of 2:
1 * 2^0 = 1
1 * 2^1 = 2
1 * 2^2 = 4
0 * 2^3 = 0
1 * 2^4 = 16
0 * 2^5 = 0
1 * 2^6 = 64
Adding these values together, we get:
1 + 2 + 4 + 16 + 64 = 87
So, 1010111 in binary is equal to 87 in decimal.
ii. To convert the number 1111011.111 to decimal, we can split it into its whole number part and fractional part.
Whole number part:
1 * 2^0 = 1
1 * 2^1 = 2
1 * 2^2 = 4
0 * 2^3 = 0
1 * 2^4 = 16
1 * 2^5 = 32
1 * 2^6 = 64
Adding these values together, we get:
1 + 2 + 4 + 16 + 32 + 64 = 119
Fractional part:
1 * 2^-1 = 0.5
1 * 2^-2 = 0.25
1 * 2^-3 = 0.125
Adding these values together, we get:
0.5 + 0.25 + 0.125 = 0.875
Combining the whole number and fractional parts, we get:
119 + 0.875 = 119.875
So, 1111011.111 in binary is equal to 119.875 in decimal.
iii. To convert the number 435.125 to binary, we can first convert the whole number part and then the fractional part.
Whole number part (435):
Divide 435 by 2 repeatedly until the quotient becomes 0:
435 / 2 = 217 remainder 1 (LSB)
217 / 2 = 108 remainder 0
108 / 2 = 54 remainder 0
54 / 2 = 27 remainder 0
27 / 2 = 13 remainder 1
13 / 2 = 6 remainder 1
6 / 2 = 3 remainder 0
3 / 2 = 1 remainder 1 (MSB)
1 / 2 = 0 remainder 1
Reading the remainders from bottom to top, we get 110110011 in binary for the whole number part.
Fractional part (0.125):
Multiply 0.125 by 2 repeatedly until reaching the desired precision:
0.125 * 2 = 0.25 (MSB)
0.25 * 2 = 0.5
Reading the resulting digits from left to right, we get 0.001 in binary for the fractional part.
Combining the whole number and fractional parts, we get:
435.12510 = 110110011.0012 in binary.
iv. To convert the number 5846 to hexadecimal, we divide by 16 repeatedly until the quotient becomes 0:
5846 / 16 = 365 remainder 6 (LSB)
365 / 16 = 22 remainder 13 (D)
22 / 16 = 1 remainder 6 (LSB)
1 / 16 = 0 remainder 1 (MSB)
Reading the remainders from bottom to top, we get 16D6 in hexadecimal for the number 5846.
i. To convert the binary number 1010111 to decimal, we use the positional notation method. The number can be expressed as:
(1 * 2^6) + (0 * 2^5) + (1 * 2^4) + (0 * 2^3) + (1 * 2^2) + (1 * 2^1) + (1 * 2^0)
Simplifying this expression, we get:
64 + 16 + 4 + 2 + 1 = 87
Therefore, 1010111 in binary is equal to 87 in decimal.
ii. To convert the binary number 1111011.111 to decimal, we also use the positional notation method. The number can be broken down as follows:
(1 * 2^6) + (1 * 2^5) + (1 * 2^4) + (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) + (1 * 2^-1) + (1 * 2^-2) + (1 * 2^-3)
Simplifying this expression, we get:
64 + 32 + 16 + 8 + 2 + 1 + 0.5 + 0.25 + 0.125 = 123.875
Therefore, 1111011.111 in binary is equal to 123.875 in decimal.
iii. To convert the decimal number 435.125 to binary, we can separate the whole number and the fractional part:
Whole number part:
Divide 435 by 2 repeatedly until the quotient becomes 0.
435 ÷ 2 = 217 remainder 1
217 ÷ 2 = 108 remainder 0
108 ÷ 2 = 54 remainder 0
54 ÷ 2 = 27 remainder 0
27 ÷ 2 = 13 remainder 1
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Reading the remainders from bottom to top, we have 110110111 in binary as the whole number part.
Fractional part:
Multiply the fractional part by 2 repeatedly until it becomes 0 or until you reach the desired precision.
0.125 * 2 = 0.25 --> Whole number part = 0
0.25 * 2 = 0.5 --> Whole number part = 0
0.5 * 2 = 1.0 --> Whole number part = 1
Reading the whole number parts from top to bottom, we have 0.001 in binary as the fractional part.
Combining both parts, we get 110110111.001 as the binary representation of 435.125.
iv. To convert the decimal number 5846 to hexadecimal, we can repeatedly divide the number by 16 until the quotient becomes 0.
5846 ÷ 16 = 365 remainder 6
365 ÷ 16 = 22 remainder 13 (represented as D in hexadecimal)
22 ÷ 16 = 1 remainder 6
1 ÷ 16 = 0 remainder 1
Reading the remainders from bottom to top, we have 16D6 in hexadecimal as the conversion of 5846.