What are binary numbers and why do we use them? BINARY NUMBERS. Converting decimal numbers to binary numbers

Transcription

1 What are binary numbers and why do we use them? BINARY NUMBERS Converting decimal numbers to binary numbers

2 The number system we commonly use is decimal numbers, also known as Base 10. Ones, tens, hundreds, and thousands. For example, 4351 represents 4 thousands, 3 hundreds, 5 tens, and 1 ones. Thousands Hundreds Tens ones

3 Thousands Hundreds Tens ones However, a computer does not understand decimal numbers. It only understands on and off, yes and no.

4 Thousands Hundreds Tens ones In order to convey yes and no to a computer, we use the numbers one ( yes or on ) and zero ( no or off ).

5 To break it down further, the number 4351 represents 1 times 1, 5 times 10, 3 times 100, and 4 times Each step to the left is another multiplication of 10. This is why it is called Base 10, or decimal numbers. The prefix decmeans ten. DECIMAL NUMBERS (BASE 10) x1000 3x100 5x10 1x1

6 One is 10 to the zero power. Anything raised to the zero power is one. Ten is 10 to the first power (or 10). One hundred is 10 to the second power (or 10 times 10). One thousand is 10 to the third power (or 10 times 10 times 10). DECIMAL NUMBERS (BASE 10) x1000 3x100 5x10 1x = = = =1

7 Binary numbers, or Base 2, use the number 2 instead of the number 10. The prefix bi- means two. Base 10 Base

8 Two raised to the zero power is one. Two raised to the first power is two. Two raised to the second power is four (or 2 times 2). Two raised to the third power is eight (or 2 times 2 times 2). Base 10 Base

9 And so on Eight times two is sixteen, or two to the fourth power. Sixteen times two is thirty-two, or two to the fifth power. Base 2 BINARY NUMBERS (BASE 2)

10 Thirty-two times two is sixty-four, or two to the sixth power. And sixty-four times two is one hundred twenty eight, or two to the seventh power. Base 2 BINARY NUMBERS (BASE 2)

11 DECIMAL 15 The number fifteen is written in decimal as one ten and five ones. In binary, the number fifteen is written as one eight, one four, one two, and one one. These are called bits, and they are either one (on) or zero (off). BINARY = =1 2 3 =8 2 2 =4 2 1 =2 2 0 =

12 8 BITS = 1 BYTE = 1 OCTET Eight bits make a byte. This is also known as an octet. When you see an IP address, it is made up of four octets (or 32 bits).

13 8 BITS = 1 BYTE = 1 OCTET If every bit is a zero that s eight zeros x x x x x x x x = = = = = = = = = 0 and we multiply each power of two by zero, and add them up the decimal equivalent of that octet is zero.

14 8 BITS = 1 BYTE = 1 OCTET x x x x x x x x = = = = = = = = = 0 x x x x x x x x = = = = = = = = = 255 If every bit is a one that s eight ones and we multiply each power of two by one, and add them up the decimal equivalent is two hundred and fifty-five. Therefore, each octet can have a value between 0 and 255.

15 Let s look at an IP address. It is easier for us to recognize decimal numbers, so we write the IP address as However, a computer sees the IP address in binary notation as four octets of ones and zeros. Decimal notation Binary notation

16 Binary notation To convert binary numbers to decimal numbers, we use the powers of two again. Write the octet below one in the 128 column, one in the sixty-four column, and zeros for the rest.

17 Binary notation = 196 Then multiply each column and add across 128 plus 64 plus zero equals 196.

18 Decimal notation 196 Binary notation = 168 Write the second octet, multiply down and add across. 128 plus 0 plus 32 plus 8 plus 0 equals 168.

19 Decimal notation Binary notation Now we ll convert the other way from decimal to binary for the third and fourth octets. To convert 131 to binary we start from the left. Can we subtract 128 from 131? Yes. So we put a one in the 128 column, and we are left with three.

20 Decimal notation Binary notation Can we subtract 64 from 3? No. So we put a zero in the 64 column. Can we subtract 32 from 3? No. Another zero for the 32 column. Zero in the 16 column, the 8 column, and the four column.

21 Decimal notation Binary notation Can we subtract a 2 from 3? Yes, and we put a one in the two column. We are left with one in the one column So 131 in binary is

22 Decimal notation Binary notation Now we ll convert the fourth octets. Starting from the left. Can we subtract 128 from 106? No. Can we subtract 64 from 106? Yes, and we are left with 42. Can we subtract 32 from 42? Yes, leaving 10.

23 Decimal notation Binary notation Can we subtract 16 from 10? No. Can we subtract 8 from 10? Yes, leaving 2. Can we subtract 4 from 2? No. Can we subtract a 2 from 2? Yes, leaving So, 106 written in binary is

24 I hope this has helped you understand a little bit about converting binary numbers. Thanks for watching! North Campus Learning Lab Room NA-113i