What is the binary representation of 35?
100011
Understanding Binary Representation of 35
The question asks for the binary representation of the decimal number 35. Binary representation is a way to express numbers using only two digits: 0 and 1. This is also known as base-2, as opposed to the decimal system, which is base-10.
To convert a decimal number, like 35, into its binary equivalent, we typically use the method of repeated division by 2. We divide the decimal number by 2 and record the remainder. We then divide the quotient by 2 and record the remainder, and so on, until the quotient becomes 0.
Step-by-Step Conversion of 35 to Binary
Here is the process of converting 35 to binary using repeated division by 2:
- Divide 35 by 2: Quotient = 17, Remainder = 1
- Divide 17 by 2: Quotient = 8, Remainder = 1
- Divide 8 by 2: Quotient = 4, Remainder = 0
- Divide 4 by 2: Quotient = 2, Remainder = 0
- Divide 2 by 2: Quotient = 1, Remainder = 0
- Divide 1 by 2: Quotient = 0, Remainder = 1
The binary representation is read from the last remainder to the first remainder (bottom-up). The remainders are 1, 0, 0, 0, 1, 1. Reading them from bottom to top gives 100011.
Let's summarize the steps in a table:
| Operation | Quotient | Remainder |
|---|---|---|
| 35 ÷ 2 | 17 | 1 |
| 17 ÷ 2 | 8 | 1 |
| 8 ÷ 2 | 4 | 0 |
| 4 ÷ 2 | 2 | 0 |
| 2 ÷ 2 | 1 | 0 |
| 1 ÷ 2 | 0 | 1 |
Reading the remainders from bottom to top (1, 0, 0, 0, 1, 1), we get the binary number 100011.
Verifying the Binary Representation 100011
We can verify this binary representation by converting it back to decimal. Each digit in a binary number represents a power of 2, starting from $2^0$ on the rightmost side.
For 100011 binary:
$$1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0$$ $$1 \times 32 + 0 \times 16 + 0 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1$$ $$32 + 0 + 0 + 0 + 2 + 1 = 35$$The decimal value is 35, which matches the original number.
Therefore, the binary representation of 35 is 100011.
Revision Table: Converting Decimal to Binary
| Concept | Description | Method for 35 |
|---|---|---|
| Decimal System | Base-10 system using digits 0-9. Each position is a power of 10. | Number 35 is in base-10. |
| Binary System | Base-2 system using digits 0-1. Each position is a power of 2. | We are converting 35 to base-2. |
| Conversion Method | Repeatedly divide the decimal number by 2 and record remainders until quotient is 0. | 35 divided by 2 repeatedly. |
| Reading Result | Read the remainders from bottom (last) to top (first). | Remainders: 1, 1, 0, 0, 0, 1 (read bottom-up as 100011). |
Additional Information: Binary Numbers in Computing
Binary representation is fundamental in computer science and digital electronics. Computers process information using electrical signals that are either on or off, which can be represented by 1 and 0 respectively. This makes the binary system the natural language for computers.
- Bits: A single binary digit (0 or 1) is called a bit.
- Bytes: A group of 8 bits is called a byte.
- Uses: Binary is used to represent data, instructions, and memory addresses within a computer.
- Signed Binary: There are different ways to represent negative numbers in binary, such as sign-magnitude, one's complement, and two's complement. Two's complement is the most common method used in modern computers.
Important Questions from Number Representations and Computer Arithmetic
Which is the largest unit of storage among the following?
The operations executed on data stored in registers are known as-
The mask logical micro-operation is equivalent to which logical gate?
- A register contains a 3's complement number 10100. If it is divided by 2, find the value of the register.
- The IEEE-754 double-precision format to represent floating point numbers, has a length of _____ bits.
