Number Systems - Binary

Base-2 (binary)

Everything in computing (hardware, software) is based on binary. At the electrical level a binary zero means “no electricity”, while a binary one means “yes electricity”.

Theory

• Uses the symbols 0 and 1 only
• Digits carry over to the next place when 1 becomes 0
• One digit can represent only two unique numbers
• Two digits can represent only four unique numbers
• Moving right to left, positions represent:
  • 2^0 = 1
  • 2^1 = 2
  • 2^2 = 4
  • 2^3 = 8

Comprehension: What number would come immediately after 1010?

Counting

In your notebook write a header Binary and create a table with two columns. In the left column put the decimal numbers 0 through 10. In the right column record the binary equivalents that you find through the counting process below.

Follow these steps/rules with your groupmates:

1. Start with all zeros across the viewer
2. Increment the rightmost strip
3. If the rightmost hits the 2, move the next strip to the left up one and move the rightmost back to 0.
4. If that second strip hits the 2, use the same method to increment the third strip and move the second back to 0.
5. Do the same for the third and fourth strips
6. Record the “output” number you have after completing 2-5, then repeat until your table is full.

Do you get it? Try each of these counts:

• Count up three decimal values from 1010
• Count up eight decimal values from 100
• Count down four decimal values from 1100
• Count down six decimal values from 1111

Conversions

Now let’s practice direct conversions in your notebook. You can choose to collaborate or do this solo:

From Binary to Decimal

To convert binary to decimal:

• Start from the right
• Multiply the digit in that place by the power of 2 corresponding to that place
• Add the results together

For example, say you have 101001:

1 * 2^0 = 1
0 * 2^1 = 0
0 * 2^2 = 0
1 * 2^3 = 8
0 * 2^4 = 0
1 * 2^5 = 32

Total = 1 + 8 + 32 = 41 in decimal

From Decimal to Binary

• Take the whole decimal number
• Divide by two
• Note the quotient and the remainder
• Divide the quotient by 2, noting the new quotient and remainder
• Repeat until your quotient reaches zero
• Record the remainders from bottom to top

For example, say you have 41:

41 / 2 = 20 remainder 1
20 / 2 = 10 remainder 0
10 / 2 = 5 remainder 0
5  / 2 = 2 remainder 1
2  / 2 = 1 remainder 0
1  / 2 = 0 remainder 1

Bottom to top it's 101001 in binary

Exercises - Conversion

1. Convert 16 decimal to binary
2. Convert 1011 binary to decimal
3. Convert 31 decimal to binary
4. Convert 10101 binary to decimal

Addition & Subtraction

You can convert back and forth to decimal and do your normal decimal addition/subtraction, but doing them right in binary is actually straight-forward.

You use the same rules as when doing addition in decimal:

• Start from the right
• Add the two digits together
• If you get zero, write it down
• If you get one, write it down
• If you get two, write a zero and carry a one to the next column

For example, let’s add 1011 and 101 like this:

  1011
+  101
------

The rightmost 1s add together to two, so a 1 gets carried and a zero written:

   (1)
  1011
+  101
------
     0

In the second column the previously carried 1 adds with the existing 1 and zero to give two. A one is carried and the zero written:

  (1)
  1011
+  101
------
    00

Again the previously carried 1 adds with the existing zero and 1 to give two. The one is carried and the zero written.

 (1)
  1011
+  101
------
   000

Finally the carried one adds with the existing one to give two, the one is carried and the zero written. Since there are no more digits for the carry, it is written too:

(1)
  1011
+  101
------
 10000

And we’re done with the result 10000.

Subtraction works the same way where you borrow from the left (so 1 becomes 10 aka two).

Exercises - Addition & Subtraction

1. Add 1010 to 101
2. Add 1010 to 1011
3. Subtract 101 from 1111
4. Subtract 11 from 1000