Structure of Binary
Binary is a place value system much like base ten, only it is based
on 2 rather than 10.
Although your students probably do not know exponents, notice there
are simple patterns they can figure out. In fact, if they only know
addition they can still figure it out:
Binary Place 
Multiplication Pattern 
Addition Pattern 
1 
1 
1 
2 
1 * 2 = 2 
1 + 1 = 2 
3 
2 * 2 = 4 
2 + 2 = 4 
4 
4 * 2 = 8 
4 + 4 = 8 
5 
8 * 2 = 16 
8 + 8 = 16 
6 
16*2 = ? 
16+16 = ? 

? * 2 = 
? + ? = 
That sequence of numbers gives us the place values for binary:
16s

8s

4s

2s

1s

Decimal Equivalent Numbers

0

0

0

1

0

2

0

0

1

0

0

4

0

0

1

1

0

4+2 = 6

0

1

0

1

0

8+2 = 10

1

0

0

0

1

16+1 = 17

Questions: What will be meant by a 1 in the sixth place? What will be
meant by a 1 in the seventh place? What is the largest number you can
write with 3 bits (places)? what is the largest you can write with 5
bits?
Patterns in Binary
If we write a sequence of binary numbers we should see some patterns
that might help us remember.
Decimal

Binary

0

000

1

001

2

010

3

011

4

100

5

101

6

110

7

111


Patterns:
[1] All odd numbers end in 1, all even numbers end in zero.
What will the last bit be for 277?
[2] The last bit (1s column) alternates: 01010101.
[3] The next column (2s place) alternates: 0 0 1 1 0 0 1
1
[4] the next column (4s place) alternates: 0 0 0 0 1 1 1
1. What do you think will be the pattern for the 8s place?

Converting From Decimal to Binary  an algorithm
We do not encourage memorizing algorithms as the correct understanding
of mathematics. However, here is an algorithm that may be discussed:
 Subtract out the highest power of 2. Put in a one to represent
that place value.
 Take your answer, and repeat step 1 until reaching 0.
 Fill in all other place values with 0, counting the 1s place as
the 0th bit.
Examples:
37

37  32 = 5

5  4 = 1

1  1 = 0

37 in binary will be: 

starting number

32 = 2^{5} > 100000

4 = 2^{2} >
100100

1 = 2^{0} > 100101

100101 
13

14  8 = 5

5  4 = 1

1  1 = 0

13 in binary will be:

starting number

14 = 2^{3} > 1000

4 = 2^{2} >
1100

1 = 2^{0} > 1101

1101

Converting Binary to Decimal  an algorithm
Simply determine the place value of each 1 and add all the place
values together:

100101> 2^{6}
+ 2^{2} + 2^{0} = 32 + 4 + 1 = 37

1101 > 2^{3}
+ 2^{2} + 2^{0} = 8 + 4 + 1 = 13

Comments:
This set of lessons was first introduced using patterns so that the
students could figure out for themselves. After that it was reinforced
with daily number practice, e.g.: Three volunteers write the day's
date in decimal, Roman numerals, and binary. This led to questions
like how long will it be before we get to use the next bit, and the
observation the the 32s place, bit #6, will never get used to write
the day of the month.
