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Learning Goals
Learning Goals
- Understand how computers count in binary
- Convert numbers between decimal and binary and back
Why Do We Count to 10?
Why Do We Count to 10?
Our number system is based on the number 10: The digits we use range form 0 to 9, a total of 10 digits, and when we run out of those 10 digits, we just add a new one to the front. This makes sense from an evolutionary standpoint because humans have 10 fingers.
But what would happen if we lived in the Simpsons world? They only have 8 fingers! Or maybe our species grew up like the Teenage Mutant Ninja Turtles, who only have 6 fingers! What would our counting system look like then? What if we were just as competent with our toes as we are with our hands, like a chimpanzee? Then we'd have 20 digits with which to count.
When a computer stores data it is using tiny electromechanical or electromagnetic switches that only have two positions: On and off, 1 or 0, true or false, yes or no. This forces a computer to think and do calculations in what we call binary. Everything is represented as numbers that are combinations of 0's and 1's.
"Traces of the anthropomorphic origin of counting systems can be found in many languages. In the Ali language (Central Africa), for example, "five" and "ten" are respectively moro and mbouna: moro is actually the word for "hand" and mbouna is a contraction of moro ("five") and bouna, meaning "two" (thus "ten"="two hands").
"Traces of the anthropomorphic origin of counting systems can be found in many languages. In the Ali language (Central Africa), for example, "five" and "ten" are respectively moro and mbouna: moro is actually the word for "hand" and mbouna is a contraction of moro ("five") and bouna, meaning "two" (thus "ten"="two hands").
It is therefore very probable that the Indo-European, Semitic and Mongolian words for the first ten numbers derive from expressions related to finger-counting. But this is an unverifiable hypothesis, since the original meanings of the names of the numbers have been lost."
It is therefore very probable that the Indo-European, Semitic and Mongolian words for the first ten numbers derive from expressions related to finger-counting. But this is an unverifiable hypothesis, since the original meanings of the names of the numbers have been lost."
- Georges Ifrah The Universal History of Numbers (Wiley, 2000, pp. 21-22)
Binary Counting
Binary Counting
In mathematics, we have been taught to think of numbers in decimal format: Each digit in a number can have a value from 0 to 9, and once we pass 9, we add a new digit.
As we count, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, we can see this pattern repeat itself. Binary is no different, except that the value of our digits can only be 0 or 1. Thus we have the table seen on the left:
As you can see, there is a pattern that emerges: Every power of 2 starts a new binary digit (1 has 1 digit, 2 has 2 digits, 4 has 3 digits, 8 has 4 digits). This trick can help us to quickly figure out how to convert binary into decimal.
Converting Binary to Decimal
Converting Binary to Decimal
Procedure:
Procedure:
- Write the powers of two in ascending order from RIGHT TO LEFT.
- Line up the binary digits with the equivalent power of 2 digit.
- Add the decimal values that correspond to the binary digits that have a 1 in the column.
Example:
Example:
Lets say we have the binary number 11010010 and we want to know what decimal number that represents, well we can think of each digit from right to left as ascending powers of 2, like this:
Using this technique, we can quickly find the value of 01101101 base 2 in decimal:
64 + 32 + 8 + 4 + 1 = 109 base 10
Converting Decimal to Binary
Converting Decimal to Binary
Converting from decimal to binary is a little more tricky, but there are 2 main methods of conversion:
Subtracting/Decomposing Powers of 2:
Subtracting/Decomposing Powers of 2:
This method goes as follows:
- Find the highest power of 2 that is smaller than your current value
- Subtract your power of 2 from the current value
- If your new value is not a power of 2, decompose your new value (back to step 1)
- If your new value is a power of 2, write a 1 for every power of 2 that you used and a 0 for every power of 2 you did not use with the largest power on the left and smallest on the right in order.
Repeated Division
Repeated Division
This method goes as follows:
- Divide your current number by 2 without creating a decimal and note the remainder (so 5 divided by 2 is not 2.5, it's 2 with 1 remainder)
- Repeat step 1 with the quotient until your quotient is 0.
- The remainder values, in reverse order of calculation give the binary value.
Practice
Practice
Try for yourself:
- Convert the following decimal numbers (base 10) to binary (base 2) using Subtracting Powers of 2:
- 12
- 38
- 89
- 158
- 221
- Verify the previous question by using Repeated Division. You should get the same binary value for each.
- Convert the following binary numbers (base 2) to decimal (base 10):
- 00001100
- 01010101
- 10101010
- 11110000
- 10010010
- 01101101
- 11111111
- 00000000
Now that we know about binary, why is it a good idea to state which base we are working in? (Hint: Think of the number 101)
See if you can convert the following values between decimal and the stated base:
- 121 base 10 to base 16 (hexadecimal)
- 27 base 10 to base 8 (octal)
- 33 base 8 (octal) to base 10
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