What are binary numbers and what is their relation to numbers we are familiar with?

Answer 1

We count in base 10. Starting with a single digit, or place value, we count out 10 digits (zero to nine), then add an additional place value on the left (making 10), and count through zero through nine again (10-19) until we have 20.

Binary numbers, however, are base 2. Instead of counting all the way to 10 in one decimal place, they only count zero to one before moving to the next place. Here's how it works for the first few binary numbers:

000: You start with zero.

001: You count to one, just like in the base ten system.

010: You count to two, but since a place value can only handle one or zero, it "overflows" into the next place value.

011: You start counting in the first place value again...

100: It overflows into the second place value again, but this time the second place value is also full, so it moves to the third place value. This is worth 4 in base 10.

This pattern continues on, with 101 (5), 110, (6), 111 (7), 1000 (8), and so on.

First Pattern: Notice that with every step, the first place value turns "on" (1) or "off" (0), following the pattern 0, 1, 0, 1, 0, 1...

With every two steps, the second place value turns "on" or "off", so it follows the pattern 0, 0, 1, 1, 0, 0, 1, 1...

With every four steps, the third place value turns "on" or "off", following the pattern 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0...

Do you see the pattern?

Second Pattern: There is one more thing you may notice about binary: Each place value is worth a power of 2. So...

First place value: Worth 0 when zero, and 1 when one

Second place value: Worth 0 when zero, and 2 when one

Third place value: Worth 4 when one

Fourth place value: Worth 8 when one

Fifth place value: Worth 16 when one

Sixth place value: Worth 32 when one...

And so on.

Example:

You want to find the "normal" (base 10) value of 11010.

A simple way to do so is to add the value of each place value that is "on". The first place is worth 1, but it is off. The second place is on, and worth 2, so add 2. The third place is off. The fourth place is on, so add 8. The fifth, and last place, is on, so add 16. You now have 2+8+16 = 26. So, 11010 is worth 26 in base 10.