Binary Number System (from http://www.mathsisfun.com/binary-number-system.html)

by techtiptom

Computers use binary digits. And some puzzles can be solved using binary numbers.

A Binary Number is made up of only 0s and 1s.

110100

Example of a Binary Number

There is no 2,3,4,5,6,7,8 or 9 in Binary!

How do we Count using Binary?

Binary

0

We start at 0

1

Then 1

???

But then there is no symbol for 2 … what do we do?

Decimal

Well how do we count in Decimal?

0

Start at 0

Count 1,2,3,4,5,6,7,8, and then…

9

This is the last digit in Decimal

10

So we start back at 0 again, but add 1 on the left

The same thing is done in binary …

Binary

0

Start at 0

1

Then 1
••

10

Now start back at 0 again, but add 1 on the left
•••

11

1 more
••••

???

But NOW what … ?

Decimal

What happens in Decimal … ?

99

When we run out of digits, we …

100

… start back at 0 again, but add 1 on the left

And that is what we do in binary …

Binary

0

Start at 0

1

Then 1
••

10

Start back at 0 again, but add 1 on the left
•••

11

••••

100

start back at 0 again, and add one to the number on the left…
… but that number is already at 1 so it also goes back to 0 …
… and 1 is added to the next position on the left
•••••

101

••••••

110

•••••••

111

••••••••

1000

Start back at 0 again (for all 3 digits),
add 1 on the left
•••••••••

1001

And so on!

Decimal vs Binary

Here are some equivalent values:

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 larger equivalent values:

Decimal:

20

25

30

40

50

100

200

500

Binary:

10100

11001

11110

101000

110010

1100100

11001000

111110100

“Binary is as easy as 1, 10, 11.”

Position

In the Decimal System there are the Units, Tens, Hundreds, etc

In Binary, there are Units, Twos, Fours, etc, like this:

This is 1×8 + 1×4 + 0×2 + 1 + 1×(1/2) + 0×(1/4) + 1×(1/8)
= 13.625 in Decimal

Numbers can be placed to the left or right of the point, to indicate values greater than one or less than one.

10.1
The number to the left of the point is a whole number (10 for example)
As we move further left, every number place
gets 2 times bigger.
The first digit on the right means halves (1/2).

As we move further right, every number place
gets 2 times smaller (half as big).

Example: 10.1

  • The “10” means 2 in decimal,
  • The “.1” means half,
  • So “10.1” in binary is 2.5 in decimal

Words

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.

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

How to Show that a Number is Binary

To show that a number is a binary number, follow it with a little 2 like this: 1012

This way people won’t think it is the decimal number “101” (one hundred and one).

Examples

Example: What is 11112 in Decimal?

  • The “1” on the left is in the “2×2×2” position, so that means 1×2×2×2 (=8)
  • The next “1” is in the “2×2” position, so that means 1×2×2 (=4)
  • The next “1” is in the “2” position, so that means 1×2 (=2)
  • The last “1” is in the units position, so that means 1
  • Answer: 1111 = 8+4+2+1 = 15 in Decimal

Example: What is 10012 in Decimal?

  • The “1” on the left is in the “2×2×2” position, so that means 1×2×2×2 (=8)
  • The “0” is in the “2×2” position, so that means 0×2×2 (=0)
  • The next “0” is in the “2” position, so that means 0×2 (=0)
  • The last “1” is in the units position, so that means 1
  • Answer: 1001 = 8+0+0+1 = 9 in Decimal

Example: What is 1.12 in Decimal?

  • The “1” on the left side is in the units position, so that means 1.
  • The 1 on the right side is in the “halves” position, so that means 1×(1/2)
  • So, 1.1 is “1 and 1 half” = 1.5 in Decimal

Example: What is 10.112 in Decimal?

  • The “1” is in the “2” position, so that means 1×2 (=2)
  • The “0” is in the units position, so that means 0
  • The “1” on the right of the point is in the “halves” position, so that means 1×(1/2)
  • The last “1” on the right side is in the “quarters” position, so that means 1×(1/4)
  • So, 10.11 is 2+0+1/2+1/4 = 2.75 in Decimal

“There are 10 kinds of people in the world,
those who understand binary numbers, and those who don’t.”

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