Binary Sequences

BINARY SEQUENCES<BR>

BINARY SEQUENCES
Chessboard

According to legend, the game of chess was invented for a Persian king by one of his servants.
The king was so pleased that he asked the servant what he would like as a reward.
The man�s request seemed very reasonable.
He asked that one grain of wheat be placed on the first square of the chessboard,
two grains on the second square, four grains on the third, and so on,
each square having twice as many grains as the square before.
The king was surprised, thinking that the servant had asked for very little.
He was even more surprised when he found out how much wheat the man actually wanted.

The numbers of grain of wheat on the squares,

1 2 4 8 16 32 64 128 ...

a geometric sequence, which grows at an increasing rate.
Because there are 64 squares on a chessboard, the inventor�s request was for as many grains of wheat as the sum of the first 64 terms of this sequence.
This number,

18,446,744,073,709,551,615

is equivalent to approximately 175 billion tons of wheat, more than that which has been produced on the earth in recorded history.

The sequence

1 2 4 8 16

is called the binary sequence.


Number  Binary Sequence

1       1   2   4   8 ...

2       1   2   4   8 ...

3       1   2   4   8 ...

4       1   2   4   8 ...

5       1   2   4   8 ...

6       1   2   4   8 ...

7       1   2   4   8 ...

8       1   2   4   8 ...

9       1   2   4   8 ...

10      1   2   4   8 ...

11      1   2   4   8 ...

12      1   2   4   8 ...

13      1   2   4   8 ...

14      1   2   4   8 ...

15      1   2   4   8 ...

The binary sequence has a remarkable property: every counting number can be expressed as the sum of one or more of its terms. This is illustrated for the numbers from 1 through 16 in the table above.

If an electric circuit consisted of a sequence of switches corresponding to the terms of the binary sequence, the counting numbers could be represented by turning the switches on or off as illustrated in the table. For example, to represent the number 3, the switches for 1 and 2 would be turned on. To represent the number 10, the switches for 2 and 8 would be turned on, and so forth.

For this reason, the binary sequence is used in the representation of numbers in the circuits of electronic computers and calculators. The circuits representing the terms of the binary sequence can be seen in this Intel 80286 Central Processing Unit.

80286 chip

To write a number in the form in which it is used in a computer, two digits are used: 1 to show that a switch is on and 0 to show that it is off. Because of this, such numbers are said to be written in base 2 or as binary numerals.

The binary numerals for the numbers from 1 to 10 are given in the table below. Note that 0�s to the left of the first 1 are customarily omitted.


Number  Binary Numeral

        8   4   2   1
1                   1

2               1   0

3               1   1

4           1   0   0

5           1   0   1

6           1   1   0

7           1   1   1

8       1   0   0   0

9       1   0   0   1

10      1   0   1   0

QUESTIONS

5.
20.
42.
71.
95.

Because every counting number can be expressed in terms of the binary sequence,
every counting number can be written as a binary numeral.
The 1's tell us to include the number and the 0�s tell us not to include the number.
Look at the examples in this table.
```
Number  Binary Numeral

    64  32  16  8   4   2   1
19           1  0   0   1   1

100  1   1   0  0   1   0   0

5    ?   ?   ?  ?   ?   ?   ?

20   ?   ?   ?  ?   ?   ?   ?  

42   ?   ?   ?  ?   ?   ?   ?

71   ?   ?   ?  ?   ?   ?   ?

95   ?   ?   ?  ?   ?   ?   ?
```
Copy the table and then add the following numbers to it, expressing each as a binary numeral.
5
20
42
71
95

The largest number that can be expressed as a three-digit binary numeral is 7 because

111 means 4 + 2 + 1 = 7

What is the largest number that can be expressed as
a four-digit binary numeral?
a five-digit binary numeral?
a six-digit binary numeral?

To change a binary numeral such as 11001 into decimal form (our usual base 10 number system), write the place value of each digit above it:
```
16  8   4   2   1
1   1   0   0   1
```
Adding the place values of the digits that are 1�s, we get

16 + 8 + 1 = 25

Use the same method to change each of the following binary numerals into decimal form:
10100
101000
1101001
11010010

Adding a zero to the end of a number in decimal form is equivalent to multiplying the number by ten.
Compare, for example, 57 and 570.
What is adding a zero to the end of a binary numeral equivalent to?

GRAINS ON A CHECKERBOARD

The number of grains of wheat said to have been requested by the inventor of chess is the sum of the first 64 terms of the binary sequence. The following figures suggest a way to find that sum without doing any addition.
Complete the following sequence from the first row of figures:

2 ? ? ? ? ?
Complete the following sequence from the second row of figures:

1 ? ? ? ? ?
How are the numbers in your answer to exercise 2 related to the numbers in your answer to exercise 1?

Copy and complete this list to show the first 12 terms of the binary sequence.


t₁  t₂  t₃  t₄  t₅  t₆  t₇ ...

1   2   4   8  16  32  64 ...

Find the sum of the first seven terms of the binary sequence by adding them:

1 + 2 + 4 + 8 + 16 + 32 + 64
Show a way to find the sum of the first seven terms without doing any addition.
What is the sum of the first 10 terms of the binary sequence?
Which term of the binary sequence would you need to know in order to quickly find the sum of the first 64 terms?

The value of that term is 18,446,744,073,709,551,616.
What is the sum of the first 64 terms of the binary sequence?

HEXADECIMAL

A long string of binary digits like 10011100110101010100011010111001
is difficult to read and comprehend as a number.
To alleviate this difficulty, computer folks invented the hexadecimal system.
Hexadecimal means 6 + 10. The hexadecimal system has 16 digits.
Hexadecimal digits combine four binary digits to make one hexadecimal digit
as is illustrated in the table below.
```
HEX   BIN    DEC
 0   0000     0

 1   0001     1

 2   0010     2

 3   0011     3

 4   0100     4

 5   0101     5

 6   0110     6

 7   0111     7

 8   1000     8

 9   1001     9

 A   1010    10

 B   1011    11

 C   1100    12

 D   1101    13

 E   1110    14

 F   1111    15
```
What are the binary AND decimal equivalents of these hexadecimal numbers?
FF₁₆
10₁₆
AA₁₆

What are the hexadecimal equivalents of these decimal numbers?
64₁₀
10₁₀
16₁₀

THE PIXEL

Change the color of the square below to YELLOW. Enter the 6 hexadecimal digits that represent YELLOW into the text field. This is how digital cameras, LCD TVs, and computer monitors work. The square represents one pixel on your TV screen. A resolution of 1280 x 800 means that more than a million pixels, like the square below, are individually encoded. The string 0000FF (or 0000ff) represents the color BLUE. You may need to experiment a little. There are only 16.7 million possibilities.
What are the 6 hexadecimal digits that represent the color YELLOW?
What are the 6 hexadecimal digits that represent the color RED?
What are the 6 hexadecimal digits that represent the color GREEN?
What are the 6 hexadecimal digits that represent the color PURPLE?
What are the 6 hexadecimal digits that represent the color ORANGE?
What are the 6 hexadecimal digits that represent the color BLACK?
What are the 6 hexadecimal digits that represent the color WHITE?

Binary Sequences

QUESTIONS

GRAINS ON A CHECKERBOARD

HEXADECIMAL

THE PIXEL