# 1.3. Binary Number System

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Since childhood, we have learned to do our computations using the numbers 0 – 9, the digits of the decimal number system. In fact, we are so accustomed to working with decimal numbers that we hardly think about their use. We balance our cheque books, pay monthly bills, and even solve algebra homework with the aid of the decimal number system. Considering the widespread use of this system, why should anyone bother to study the binary number system? The answer is found in something that is almost as widespread as decimal numbers: computers.

The binary system is the number system based on 2 symbols. Computer storage uses different means of recording information based on a series of electronic components that are able to hold a charge. At any time the component can be in one of two states, charged or not charged. Any data stored in a computer is stored as a series of binary values whether it is text, numbers, pictures, sound or video.

While it is fine for us to use ten digits for our computations, computers do not have this luxury. Every computer processor is made of millions of tiny switches that can be turned off or on. Since these switches only have two states, it makes sense for a computer to perform its computations with a number system that only has two digits: the binary number system. These digits (0 and 1) are called bits and correspond to the off/on positions of the switches in the computer processor. With only these two digits, a computer can perform all the arithmetic that we can with ten digits.

Our study of the binary system will help us gain a better understanding of how computers perform computations. In a way, we can think of this study as learning another language, the language of the computer. Every instruction that a computer executes is coded in this binary language.

The binary number system uses TWO values to represent numbers. The values are:

0 and 1

with 0 having the least value, and 1 having the greatest value. Columns are used in the same way as in the decimal system, in that the left most column is used to represent the greatest value.

In a computer, a binary variable capable of storing a binary value (0 or 1) is called a BIT. A series of eight bits strung together makes a byte. With 8 bits, or 8 binary digits, there exist 2^8=256 possible combinations.

To understand binary numbers, begin by recalling primary school math. When we first learned about numbers, we were taught that, in the decimal system, things are organized into columns:

H | T | O

1 | 9 | 3

such that “H” is the hundreds column, “T” is the tens column, and “O” is the ones column. So the number “193” is 1-hundreds plus 9-tens plus 3-ones.

Years later, we learned that the ones column meant 10^0, the tens column meant 10^1, the hundreds column 10^2 and so on, such that

10^2|10^1|10^0

1 | 9 | 3

the number 193 is really {(1*10^2)+(9*10^1)+(3*10^0)}.

As you know, the decimal system uses the digits 0-9 to represent numbers.

Power of the base | 10^{4} | 10^{3} | 10^{2} | 10^{1} | 10^{0} |

Position value | 10,000 | 1000 | 100 | 10 | 1 |

If we wanted to put a larger number in column 10^n (e.g., 10), we would have to multiply 10*10^n, which would give 10^(n+1), and be carried a column to the left. For example, putting ten in the 10^0 column is impossible, so we put a 1 in the 10^1 column, and a 0 in the 10^0 column, thus using two columns. Twelve would be 12*10^0, or 10^0(10+2), or 10^1+2*10^0, which also uses an additional column to the left (12).

The binary system works under the exact same principles as the decimal system, only it operates in base 2 rather than base 10. In other words, instead of columns being

10^2|10^1|10^0

they are

2^2|2^1|2^0

Instead of using the digits 0-9, we only use 0-1 (again, if we used anything larger it would be like multiplying 2*2^n and getting 2^n+1, which would not fit in the 2^n column. Therefore, it would shift you one column to the left. For example, “3” in binary cannot be put into one column. The first column we fill is the right-most column, which is 2^0, or 1. Since 3>1, we need to use an extra column to the left, and indicate it as “11” in binary (1*2^1) + (1*2^0).

Power of the base | 2^{4} | 2^{3} | 2^{2} | 2^{1} | 2^{0} |

Positional value | 16 | 8 | 4 | 2 | 1 |

The **word binary** comes from “Bi-” meaning two. We see “bi-” in words such as “bicycle” (two wheels) or “binocular” (two eyes).

**When you say a binary number**, pronounce each digit (example, the binary number “101” is spoken as “one zero one”, or sometimes “one-oh-one”). This way people don’t get confused with the decimal number.

Because you can only have 0s or 1s, this is how you count using Binary:

Decimal: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

Binary: | 0 | 1 | 10 | 11 | 100 | 101 | 110 | 111 | 1000 | 1001 | 1010 | 1011 | 1100 | 1101 | 1110 | 1111 |

Here are some more equivalent values:

Decimal: | 20 | 25 | 30 | 40 | 50 | 100 | 200 | 500 |

Binary: | 10100 | 11001 | 11110 | 101000 | 110010 | 1100100 | 11001000 | 111110100 |

A binary digit is called a bit:

- A single binary digit (like “0” or “1”) is called a “bit”. For example 11010 is five bits long.
- The word bit is made up from the words “binary digit”
- To show that a number is a binary number, follow it with a little 2 like this: 101
_{2 }This way people won’t think it is the decimal number “101” (one hundred and one). - There are two possible states in a bit, Usually expressed as 0 and 1, the two numbers used in the binary number system.

But the bit could represent on / off of an electrical circuit, yes / no, true / false, -1 / 0, -1 / +1, zero / non zero, or similar 2 state binary wording.

**Examples**:

- What would the binary number 1011 be in decimal notation?

1011 =

—- 1 X 2^{0} = 1

—– 1 X 2^{1} = 2

—— 0 X 2^{2} = 0

——- 1 X 2^{3} = 8

—-

11 (in decimal)

- When the decimal number 4635 is written into the columns you can see that its value is:

position 4 | position 3 | position 2 | position 1 |

10^{3} | 10^{2} | 10^{1} | 10^{0} |

1000 | 100 | 10 | 1 |

4 | 6 | 3 | 5 |

= 4 * 1000 + 6 * 100 + 3 * 10 + 5 * 1

= 4000 + 600 + 30 + 5

- Another example of a binary number