Binary Number Conversions

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The binary system is important in computer technology since any binary number can be represented by, for example, the positions of a series of on-off switches. The "on" position corresponds to a 1, and the "off" position to a 0. Instead of switches, magnetized dots on a magnetic tape or disk also can be used to represent binary numbers: a magnetized dot stands for the digit 1, and the absence of a magnetized dot is the digit 0. Flip-flops-electronic devices that can only carry two distinct voltages at their outputs and that can be switched from one state to the other state by an impulse-can also be used to represent binary numbers; the two voltages correspond to the two digits.

Any decimal number can be expressed in the binary system by the sum of different powers of two.

Take for example: 101 and convert it to decimal

Start from the right

1 x 2^0 + 0 x 2 ^1 + 1 x 2^2 = 1 x 1 + 0 x 2 + 1 x 4 = 1 + 0 + 4 = 5

Another example, the conversion of the binary number 10101101 into a decimal number.

Start from the right and

(1 × 2^0) + (0 × 2^1) + (1 × 2^2) + (1 × 2^3) + (0 × 2^4) + (1 × 2^5) + (0 × 2^6) + (1 × 2^7) = 173.

The following table (8-4-2-1) can be created: Showing the relationship between base 10 and base 2 numbers. Base 10 (decimal) numbers are arranged as powers of 10

10^0 = 1 (ones), 10^1 = 10 (tens), 10^2 = 100 (hundreds), 10^3 = 1000 (thousands).

Base 2 (binary) numbers are arranged as powers of 2.

2^0 = 1 (ones), 2^1 = 2 (twos), 2^2 = 4 (fours), 2^3 = 8 (eights)

 8-4-2-1 Table 8 = 2^3 4 = 2^2 2 = 2^1 1 = 2^0 0 = 10^0 0 0 0 0 1 0 0 0 1 2 0 0 1 0 3 0 0 1 1 4 0 1 0 0 5 0 1 0 1 6 0 1 1 0 7 0 1 1 1 8 1 0 0 0 9 1 0 0 1 10 = 10^1 1 0 1 0 11 1 0 1 1 12 1 1 0 0 13 1 1 0 1 14 1 1 1 0 15 1 1 1 1

For the conversion of decimal numbers to binary numbers, the same principle can be used, but the other way around.

To convert 15 to a binary number look for the highest power of 2 that is less than or equal to the decimal number in this case 2^4 = 16 (too much) but 2^3 = 8 so 1 can be inserted in the fourth place from the right. The remainder is 7 after subtracting the 8 so we find 2^2=4 which is less than 7 with 7-4=3, so we have a 1 marked in the third place from the right. 2^1 is less than 3 with 3-2=1 so we have a 1 marked in the second position from the right. With one left over all we have left is 2^0=1 so we have 1 marked in the far right position. Thus we have 15 = 1111.

Try 64 converted to binary (hint: 8*8 = 64, and 2^3 = 8)

Arithmetic operations in the binary system are extremely simple. The basic rules are: 1 + 1 = 10, and 1 × 1 = 1. Zero plays its usual role: 1 × 0 = 0, and 1 + 0 = 1. Addition, subtraction, and multiplication are done in a fashion similar to that of the decimal system.

#### Use this converter when you have learned why and how to convert. (Hint: Press the tab button after entering the number and the base to convert)

Converted to Base =

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