Convert (513) 10 Decimal number to (..) 2 Binary number.

To convert the decimal number $(513{)}_{10}$ to its binary equivalent, we use the division-by-2 method. This involves repeatedly dividing the number by 2 and recording the remainders. The binary number is obtained by reading the remainders in reverse order.

Step-by-Step Conversion:

1. Divide 513 by 2: $$513÷2=256\text{with a remainder of }1$$

2. Divide 256 by 2: $$256÷2=128\text{with a remainder of }0$$

3. Divide 128 by 2: $$128÷2=64\text{with a remainder of }0$$

4. Divide 64 by 2: $$64÷2=32\text{with a remainder of }0$$

5. Divide 32 by 2: $$32÷2=16\text{with a remainder of }0$$

6. Divide 16 by 2: $$16÷2=8\text{with a remainder of }0$$

7. Divide 8 by 2: $$8÷2=4\text{with a remainder of }0$$

8. Divide 4 by 2: $$4÷2=2\text{with a remainder of }0$$

9. Divide 2 by 2: $$2÷2=1\text{with a remainder of }0$$

10. Divide 1 by 2: $$1÷2=0\text{with a remainder of }1$$

Now, we write the remainders in reverse order (from the last division to the first):

$$(513{)}_{10}=(1000000001{)}_2$$

Verification:

To ensure correctness, we can convert $(1000000001{)}_2$ back to decimal: $$1\times 2^9+0\times 2^8+0\times 2^7+0\times 2^6+0\times 2^5+0\times 2^4+0\times 2^3+0\times 2^2+0\times 2^1+1\times 2^0$$ $$=512+1=513$$

Thus, the binary equivalent of $(513{)}_{10}$ is $(1000000001{)}_2$.