Numbering System with MCQ'S

Numbering Systems

The binary and hexidecimal numbering systems have always presented a huge problem for most students. Actually, if I look through the different computer books and study material that try to explain the subject, its really no wonder. I have found information that explains it using boxes, building blocks, candles, cars.... I have no idea what this is myself. In fact, just as you start to think you've got a handle on it, you read another explanation that tends to confuse you entirely. So, at the risk of baffling you completely, I'll try to explain, using another method, the method my grade 2 teacher (or maybe grade 3, I can't remember) taught me to understand decimal numbers of more than 1 digit. This doesn't work so well for hexidecimal, because the numbers get real large real quick. But we can check it out.

First thing to remember... is that you don't need to learn to count. Unless you're attempting to understand low level programming, learning binary provides you with two things:
- comprehension of how your computer stores information, with a better understanding of bits, bytes, kilobytes, etc...
- an easy method of converting between decimal and hexidecimal. You don't have to learn to count in hexidecimal either, but I/O addresses and memory addresses are given in hex and sometimes you can resolve device conflicts by being able to tell if addresses overlap.

So, my suggestion is to find a conversion method that you can understand, bookmark it or keep it in a file, then forget about it. Look it up again on the rare occasion that you might need it. (By the way, my favorite conversion method is a calculator).

All that being said, lets get back to grade 2.

Decimal (base 10)

The decimal numbering system is the system we all use in our everyday work. It consists of 10 digits (hence the name decimal). The digits, from smallest to largest are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Now, any number can be broken down into columns. From right to left, the first column is the 1's column, then the 10's column, then the 100's, then the 1000's, etc..
Let's look at the number 137:

Thousands	Hundreds	Tens	Ones
0	1	3	7

0 X 1000 = 0 (0 thousands)
1 X 100 = 100 (1 hundreds)
3 X 10 = 30 (3 tens)
7 X 1 = 7 (7 ones)

0+100 + 30 + 7 = 137

If you don't understand it this far, go back over it until you do, or get some help.

OK, lets try the number 77:

Thousands	Hundreds	Tens	Ones
0	0	7	7

0 X 1000 = 0 (0 thousands)
0 X 100 = 0 (0 hundreds)
7 X 10 = 70 (7 tens)
7 X 1 = 7 (7 ones)

0+0+70 + 7 = 77

Binary (base 2)

Your computer works using the binary numbering system. This is because a computer can only recognize two states, the presence of an electrical charge or the absence of an electrical charge. In other words, on or off.

The binary numbering system is ideal for representing these two states because it consists of only two digits. The digits, from smallest to largest are:
0, 1.

0 represents the absence of an electrical charge or 'off'.
1 represents the presence of an electrical charge or 'on'.
(Now you can understand why the power switch on your computer is labelled as O/1).

Once again, any number can be broken down into columns. Each column is a placeholder. Using the binary numbering system, from right to left, the first column is the 1's column, then the 2's column, then the 4's, then the 8's, then the 16's column, then the 32's column, etc..
Let's look at the number 137 again:

One hundred twenty-eights	Sixty- fours	Thirty- twos	Sixteens	Eights	Fours	Twos	Ones
1	0	0	0	1	0	0	1

1 X 128 = 128 (1 one hundred twenty-eights)
0 X 64 = 0 (0 sixty-fours)
0 X 32 = 0 (0 thirty-twos)
0 X 16 =0 (0 sixteens)
1 X 8 = 8 (1 eights)
0 X 4 = 0 (0 fours)
0 X 2 = 0 (0 twos)
1 X 1 = 1 (1 ones)

128 + 0 + 0 + 0 + 8 + 0 + 0 + 1 = 137
Thus, the binary number 10001001 is equal to 137 decimal.

A single digit (0 or 1) is called a 'bit' (binary digit).
The table above contains 8 bits. Each column can contain either a 1 or a 0 ( 'cause there is only 2 digits in the binary numbering system). So, as you can see, it takes 8 bits to represent the decimal number 137.

Make sense?
If you don't understand it this far, go back over it until you do, or get some help.

OK, lets try the number 77:

One hundred twenty-eights	Sixty- fours	Thirty- twos	Sixteens	Eights	Fours	Twos	Ones
0	1	0	0	1	1	0	1

0 X 128 = 0 (0 one hundred twenty-eights)
1 X 64 = 64 (1 sixty-fours)
0 X 32 = 0 (0 thirty-twos)
0 X 16 =0 (0 sixteens)
1 X 8 = 8 (1 eights)
1 X 4 = 4 (1 fours)
0 X 2 = 0 (0 twos)
1 X 1 = 1 (1 ones)

0 + 64 + 0 + 0 + 8 + 4 + 0 + 1 = 77
Thus, the binary number 01001101 is equal to 77 decimal.

Any character that you type (including spaces), and any character you can see on your screen, takes 8 bits for your computer to produce.Eight bits associated together are called a 'byte'. A byte is the real building block of computer information. This can form a single character. A character can be a number, letter, or symbol. The amount of information a device can store is measured in bytes. Because of this, most numbers and sizes reported by the computer are evenly divisible by 8.
Multiple bytes associated together are often called a 'word'.

8 bits = 1 byte
1024 bytes = 1 kilobyte (Kb)
1024 kilobytes = 1 megabyte (Mb) (1,048,576 characters or bytes)
1024 megabytes = 1 gigabyte (Gb)

These numbers are often rounded off for ease of calculation.
You might see 1 megabyte represented as a million bytes (1,000,000 characters), or 1000 kilobytes.

Hexidecimal (base 16) Remember, that eight bits make a byte, and that it takes eight bits to store a single character in the computer. 1024 bytes make a kilobyte and 1024 is divisible by eight. Seems like the magic number for computers is eight. The thing is, even relatively low binary numbers can be long and cumbersome. The hexidecimal numbering system consists of sixteen digits. The digits, from smallest to largest are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Once again, any number can be broken down into columns. Each column is a placeholder. Using the hexidecimal numbering system, from right to left, the first column is the 1's column, then the 16's column, then the 256's, then the 4096's, then the 65536's column, etc.. Because you multiply the number in each previous column by 16, you can see that the numbers get very large and scary very quickly. Here's the number 137: 65536 4096 256 16 1 8 9 0 X 65536 = 0 (0 sixty-five thousand five hundred thirty-sixes) 0 X 4096 = 0 (0 four thousand ninety-sixes) 8 X 16 = 128 9 X 1 = 9 0 + 0 + 128 + 9 = 137 Decimal 137 = 89h (Hexidecimal) Here's an easier way. Any group of 4 binary digits can represent up to sixteen values. A single hexidecimal digit can stand for any one of the 16 values that can be represented by any 4 bit binary number. In other words, break any binary number into groups of 4 bits. Use a chart (or if your memory is good you can use that) to find the proper hex number that represents those four bits. I know, it sounds a bit confusing, so let's do a couple. Here's the chart: Decimal Hex Binary 0 0 0000 1 1 0001 2 2 0010 3 3 0011 4 4 0100 5 5 0101 6 6 0110 7 7 0111 8 8 1000 9 9 1001 10 A 1010 11 B 1011 12 C 1100 13 D 1101 14 E 1110 15 F 1111 Lets try the number 137 again: From the previous page, we know that the binary equivalent is 10001001. Break this into 2 groups of 4 bits, [1000][1001] Use the chart to find Hex numbers [1000] = 8h [1001] = 9h Decimal = 137 Binary equivalent = 10001001 Hexidecimal equivalent = 89h Make sense? If you don't understand it this far, go back over it until you do, or get some help. Hexidecimal numbers are followed by a lowercase 'h' to designate them as hex. One thing to keep in mind is that if you have a binary number like 10 (2 decimal), you can add zeros to the left to make it a full 4 bit number. eg. 0010 Decimal Hex Binary 0 0 0000 1 1 0001 2 2 0010 3 3 0011 4 4 0100 5 5 0101 6 6 0110 7 7 0111 8 8 1000 9 9 1001 10 A 1010 11 B 1011 12 C 1100 13 D 1101 14 E 1110 15 F 1111 OK, lets try the number 77: We know from the previous page that the binary equivalent is 01001101. Let's break that into two groups of 4 bits. [0100][1101]. Now let's check the chart. [0100] = 4h [1101] = Dh Decimal = 77 Binary equivalent = 01001101 Hexidecimal equivalent = 4Dh Check it! 65536 4096 256 16 1 4 D 0 X 65536 = 0 (0 sixty-five thousand five hundred thirty-sixes) 0 X 4096 = 0 (0 four thousand ninety-sixes) 0 X 256 = 0 (0 two hundred fifty-sixes) 4 X 16 = 64 (4 sixteens) D X 1 = 13 (D ones) (remember we're using hex, D = 13 decimal) 0 +0 + 0 + 64 + 13 = 77 decimal Now, if all this has completely confused you, don't worry. Remember, this isn't your only resource. Check out your books, the internet, ask someone else. Once you've found a method that you can understand, keep it in mind (or maybe a safer place). If you need to convert a number, refer to it, make the conversion, then lock it away again.

Decimal

100,000

10,000

1,000

100

10

1

Binary

512

256

128

64

32

16

8

4

2

1

Hexidecimal

65,536

4,096

256

16

1

You didn't think you were getting away that easy, did you? You have to do a little practise before you can lock it away entirely!
The digit values are above and your chart is down below. See if you can fill in the missing values.

Decimal	Hex	Binary
0	0	0000
1	1	0001
2	2	0010
3	3	0011
4	4	0100
5	5	0101
6	6	0110
7	7	0111
8	8	1000
9	9	1001
10	A	1010
11	B	1011
12	C	1100
13	D	1101
14	E	1110
15	F	1111

Decimal	Hex			Binary
163	?			10100011
?	2Fh			?
?	30h			00110000
3840	?			?
?	?			10000000000
?	4ACh			?
Decimal		Hex	Binary
163		A3h	10100011
47		2Fh	00101111
48		30h	00110000
3840		F00h	111100000000
1024		400h	10000000000
1196		4ACh	010010101100

Binary & Hexidecimal - Review 1. One gigabyte is equal to ______________ . a) 1024 kilobytes b) 1024 megabytes c) 1024 bytes d) 8 bits 2. The decimal number 64 can be represented by which binary number? a) 00100000 b) 10100000 c) 01000000 d) 11000010 3. What is the largest decimal number that can be represented using an 8-bit binary number? a) 255 b) 1024 c) 256 d) 4096 4. The binary number 00001100 can be represented by which hexidecimal number? a) C h b) D h c) 4C h d) A1 h 5. True or False? The binary number 1101 is equal to 13 decimal. a) True b) False 6. The decimal number 15 is equal to the hexidecimal number Fh. What is the hexidecimal equivalent of decimal 16? a) Gh b) 1h c) 1Fh d) 10h 7. True or False? Hexidecimal numbers are always followed by a lowercase 'h' to designate them as hex. a) True b) False