The binary number 10101 is equivalent to the decimal number
- This is determined by calculating the positional values of each digit in the binary number, which are based on powers of 2.
Understanding the difference between binary and decimal
To understand how the binary number 10101 is converted to 21, it's essential to first grasp the difference between the decimal and binary number systems.
The decimal (base-10) system
The decimal system is the standard numbering system we use every day. It is a base-10 system, meaning it uses ten digits (0–9) to represent numbers. The position of each digit in a decimal number determines its value as a power of 10. For example, in the decimal number 123:
-
The 3 is in the "ones" place, so its value is 3×100=33 cross 10 to the 0 power equals 3
3×100=3
.
-
The 2 is in the "tens" place, so its value is 2×101=202 cross 10 to the first power equals 20
2×101=20
.
-
The 1 is in the "hundreds" place, so its value is 1×102=1001 cross 10 squared equals 100
1×102=100
.The total value is the sum of these positions: 100+20+3=123100 plus 20 plus 3 equals 123
100+20+3=123
.
The binary (base-2) system
The binary system, on the other hand, is a base-2 system that uses only two digits, 0 and 1. It is the fundamental language of all digital computers. The position of each digit in a binary number (called a "bit") determines its value as a power of 2. This is how computers efficiently store and process data, representing "on" and "off" states of electrical circuits.
Step-by-step conversion of 10101
To convert the binary number 10101 to its decimal equivalent, follow these steps:
**1. Write down the binary number:**10101
**2. Assign a positional value to each digit:**Start from the rightmost digit and move left, assigning each digit a power of 2, starting with 202 to the 0 power
20
.
-
1 (rightmost digit) is in the 202 to the 0 power
20
position.
-
0 is in the 212 to the first power
21
position.
-
1 is in the 222 squared
22
position.
-
0 is in the 232 cubed
23
position.
-
1 (leftmost digit) is in the 242 to the fourth power
24
position.
**3. Multiply each digit by its positional value:**Multiply each binary digit by its corresponding power of 2.
-
1×24=1×16=161 cross 2 to the fourth power equals 1 cross 16 equals 16
1×24=1×16=16
-
0×23=0×8=00 cross 2 cubed equals 0 cross 8 equals 0
0×23=0×8=0
-
1×22=1×4=41 cross 2 squared equals 1 cross 4 equals 4
1×22=1×4=4
-
0×21=0×2=00 cross 2 to the first power equals 0 cross 2 equals 0
0×21=0×2=0
-
1×20=1×1=11 cross 2 to the 0 power equals 1 cross 1 equals 1
1×20=1×1=1
**4. Sum the results:**Add up all the products from the previous step.
16+0+4+0+1=2116 plus 0 plus 4 plus 0 plus 1 equals 21
16+0+4+0+1=21
The final result confirms that the binary number 10101 is equivalent to the decimal number 21.