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

To convert the decimal number $(513{)}_{10}$ to its binary equivalent $(..{)}_2$, we use the division-by-2 method. This method involves repeatedly dividing the number by 2 and recording the remainders, which form the binary digits (bits) in reverse order.

Step-by-Step Conversion:

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

2. Divide the quotient (256) by 2 and record the remainder: $$256÷2=128\text{with a remainder of }0$$

3. Divide the quotient (128) by 2 and record the remainder: $$128÷2=64\text{with a remainder of }0$$

4. Divide the quotient (64) by 2 and record the remainder: $$64÷2=32\text{with a remainder of }0$$

5. Divide the quotient (32) by 2 and record the remainder: $$32÷2=16\text{with a remainder of }0$$

6. Divide the quotient (16) by 2 and record the remainder: $$16÷2=8\text{with a remainder of }0$$

7. Divide the quotient (8) by 2 and record the remainder: $$8÷2=4\text{with a remainder of }0$$

8. Divide the quotient (4) by 2 and record the remainder: $$4÷2=2\text{with a remainder of }0$$

9. Divide the quotient (2) by 2 and record the remainder: $$2÷2=1\text{with a remainder of }0$$

10. Divide the quotient (1) by 2 and record the remainder: $$1÷2=0\text{with a remainder of }1$$

Writing the Binary Number:

Now, write the remainders in reverse order (from the last division to the first). This gives us the binary representation of $(513{)}_{10}$:

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

Verification:

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

Thus, the conversion is correct.

Final Answer: $$(513{)}_{10}=\boxed{(1000000001{)}_2}$$