BASE NUMBER SYSTEMS:BINARY AND TERNARY NUMBERS

Honors Precalculus

Mr. Velazquez

OUR SYSTEM: BASE 10When we express numbers, we do so using ten numerical characters which cycle every multiple of 10. The reason for this is simple: we have 10 fingers

This is not the only way of expressing numbers; this is simply one particular way that we’ve decided to use for convention

If we had more or less than 10 fingers, our number system might look very different.

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

Next digit

changes for every

multiple of 10

OUR SYSTEM: BASE 10A benefit of expressing numbers as a scrolling series of symbols is that we can express any number as a series of powers of the base.

In number theory, a branch of mathematics that deals with the properties and relationships of numbers, we envision each number as the value of a polynomial function with positive integer coefficients, and with 𝑥 equal to the base.

For example: 𝟏𝟑 = 1 10 1 + 3 10 0 = 10+ 3

𝟓𝟕𝟏 = 5 10 2 + 7 10 1 + 1 10 0 = 500+ 70 + 1

𝟐, 𝟎𝟗𝟔 = 2 10 4 + 0 10 3 + 9 10 1 + 6 10 0 = 2,000+ 90 +6

Note that the 100 terms here

are the constant terms

(because 100 = 𝑛0 = 1)

BASE REPRESENTATION OF NUMBERS

Any positive whole number 𝑘 can be expressed as the sum of a series of powers of any natural number base 𝑏.

𝑘 = 𝑎𝑛𝑏𝑛+ 𝑎𝑛−1𝑏

𝑛−1+⋯+ 𝑎2𝑏2+ 𝑎1𝑏 + 𝑎0

Where 𝑎𝑛, 𝑎𝑛−1,… , 𝑎2, 𝑎1 and 𝑎0 are all equal or greater than zero and are the numeric digits of 𝑘 in base 𝑏.

Much of our basic arithmetic relies on this simple fact. For instance:

212 + 573 = 2 10 2 + 1 10 + 2 + [5 10 2 + 7 10 + 3]212 + 573 = 200 + 500 + 10 + 70 + 2+ 3213 + 573 = 700 + 80 + 5 = 𝟕𝟖𝟓

BASE 2: BINARY NUMBERS

The simplest form of numeral system is binary, which uses a base of 2, and thus only requires 2 different symbols to express numbers. The symbols we use are 0 and 1.

This system is ideal for computers, since they can only sense when a switch is ON (1) or OFF (0).

(Hope you remember

your powers of two!)

BASE 2: CONVERTING TO BINARY

Step 0: Write out the powers of two from right to left (you don’t have to write the full number, just use 20, 21, etc.) then choose your number to convert, which we will call 𝑥

Example: Let’s suppose 𝑥 = 764

Step 1: From your list of powers of 2, find the highest power of 2 that is less than your number 𝑥, and circle that power

Example: The highest power of 2 that is less than 764 is 29 = 512, so circle 29 on the list

Step 2: Subtract that power of 2 from your number 𝑥. You will be left with a remainder, which we will call 𝑟1

Example: 𝑟1 = 764 − 512 = 252

BASE 2: CONVERTING TO BINARYStep 3: From your list of powers of 2, find the highest power of two that is less than your remainder 𝑟1 , and circle that power

Example: The highest power of 2 less than 252 is 27 = 128, so circle 27 on the list

Step 4: Repeat steps 2 and 3 for remainder 𝑟1 , circling and subtracting the largest possible powers of 2 until you have no more remainder (don’t forget to circle the powers that you are subtracting!)

Example:

𝑟2 = 𝑟1 − 27 = 252 − 128 = 124𝑟3 = 𝑟2 − 26 = 124 −64 = 60𝑟4 = 𝑟3 − 25 = 60 −32 = 28𝑟5 = 𝑟4 − 24 = 28 −16 = 12𝑟6 = 𝑟5 − 23 = 12 − 8 = 4𝑟7 = 𝑟6 − 22 = 4 − 4 = 0

BASE 2: CONVERTING TO BINARY

Step 5: You will now have a list of descending powers of 2, with some circled and some not circled. All you need to do now is write a “1” above each circled power and a “0” above each power that’s not circled. (You do not need to do this for any powers greater than the highest one you used)

Example: We used the powers 9, 7, 6, 5, 4, 3, and 2, but did not use 8, 1 and 0

1

|

0

|

1

|

1

|

1

|

1

|

1

|

1

|

0

|

0

|

29 28 27 26 25 24 23 22 21 20

Step 6: This series of 1’s and 0’s is your number 𝑥, in binary form.

Example: 76410 = 10111111002 or alternatively, DEC 764 = BIN 1011111100

BASE 2: CONVERTING TO BINARYConvert the following numbers into binary:

52

433

BASE 2: CONVERTING FROM BINARY

BASE 2: CONVERTING FROM BINARY

BASE 2: A FEW MORE THINGS

COUNTING BINARY ON YOUR FINGERSBy assigning a power of two to each finger, and using a system where each extended finger represents a “1” in binary, we can use our fingers to count all the way to 1023.

COUNTING BINARY ON YOUR FINGERSBy assigning a power of two to each finger, and using a system where each extended finger represents a “1” in binary, we can use our fingers to count all the way to 1023.

USING BINARY TO FOOL PEOPLE

You can use special cards like these to convince people that you can read their minds.

CLASSWORK PART 1 (BINARY)

BASE 3: TERNARY NUMBERS

BASE 3: TERNARY NUMBERS

BASE 3: CONVERTING TO DECIMAL

USING TERNARY TO FOOL PEOPLE

Then of course, there’s the base-3 card trick

It takes time and practice to get this trick right. Here’s a link to an explanation by mathematician Matt Parker on how to do it properly: https://www.youtube.com/watch?v=l7lP9y7Bb5g

CLASSWORK PART 2 (TERNARY)