Useful Number Systems

Decimal Base = 10

Digit Set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Binary Base = 2

Digit Set = {0, 1}

Octal Base = 8 = 2³

Digit Set = {0, 1, 2, 3, 4, 5, 6, 7}

Hexadecimal Base = 16 = 2⁴

Digit Set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}

Common notation:

Leading 0 denotes octal 077 is an octal number, same as decimal 63

Leading 0x denotes hexadecimal 0x77 is hexadecimal (decimal 119)

Binary, Decimal, and Hexadecimal Equivalents

Binary Decimal Hexadecimal

0000 0 0

0001 1 1

0010 2 2

0011 3 3

0100 4 4

0101 5 5

0110 6 6

0111 7 7

1000 8 8

1001 9 9

1010 10 A

1011 11 B

1100 12 C

1101 13 D

1110 14 E

1111 15 F

Conversion Between Binary and Hexadecimal

This is easy, just group the bits. Recall that
A = 1010 B = 1011 C = 1100
D = 1101 E = 1110 F = 1111

Problem: Convert 10011100 to hexadecimal.
1. Group by fours 1001 1100
2. Convert each group of four 0x9C

Problem: Convert1111010111 to hexadecimal.
1. Group by fours (moving right to left) 11 1101 0111
Group by fours 0011 1101 0111
2. Convert each group of four 0x3D7

Problem: Convert 0xBAD1 to binary
1. Convert each hexadecimal digit: B A D 1
1011 1010 1101 0001

2. Group the binary bits 1011101011010001

Conversion Between Binary and Decimal

Conversion between hexadecimal and binary is easy because 16 = 2⁴.
In my book, hexadecimal is just a convenient “shorthand” for binary.
Thus, four hex digits stand for 16 bits, 8 hex digits for 32 bits, etc.

But 10 is not a power of 2, so we must use different methods.

Conversion from Binary to Decimal

This is based on standard positional notation.

Convert each “position” to its decimal equivalent and add them up.

Conversion from Decimal to Binary

This is done with two distinct algorithms, one for the digits to the left of
the decimal point (the whole number part) and one for digits to the right.

At this point we ignore negative numbers.

Powers of Two

Students should memorize the first ten powers of two.

2⁰ = 1
2¹ = 2 2^–1 = 0.5
2² = 4 2^–2 = 0.25
2³ = 8 2^–3 = 0.125

2⁴ = 16 2^–4 = 0.0625
2⁵ = 32 2^–5 = 0.03125
2⁶ = 64 etc.
2⁷ = 128
2⁸ = 256
2⁹ = 512
2¹⁰ = 1024

10111.011 = 1·2⁴ + 0·2³ + 1·2² + 1·2¹ + 1·2⁰ + 0·2^-1 + 1·2^-2 + 1·2^-3

= 1·16 + 0·8 + 1·4 + 1·2 + 1·1 + 0·0.5 + 1·0.25 + 1·0.125
= 23.375

Conversion of Unsigned Decimal to Binary

Again, we continue to ignore negative numbers.

Problem: Convert 23.375 to binary. We already know the answer.

One solution.

23.375 = 16 + 4 + 2 + 1 + 0.25 + 0.125
= 1·2⁴ + 0·2³ + 1·2² + 1·2¹ + 1·2⁰ + 0·2^-1 + 1·2^-2 + 1·2^-3
= 10111.011

This solution is preferred by your instructor, but most students find it
confusing and prefer to use the method to be discussed next.

Side point: Conversion of the above to hexadecimal involves grouping
the bits by fours as follows:
Left of decimal: by fours from the right
Right of decimal: by fours from the left.

Thus the number is 00010111.0110, or 0001 0111.0110 or 0x17.6

But 0x17.6 = 1·16 + 7·1 + 6/16 = 23 + 3/8 = 23.375

Conversion of the “Whole Number” Part

This is done by repeated division, with the remainders forming the binary
number. This set of remainders is read “bottom to top”

Quotient Remainder

23/2 = 11 1 Thus decimal 23 = binary 10111

11/2 = 5 1

5/2 = 2 1 Remember to read the binary

2/2 = 1 0 number from bottom to top.

1/2 = 0 1 As expected, the number is 10111

Another example: 16

Quotient Remainder

16/2 = 8 0

8/2 = 4 0

4/2 = 2 0 Remember to read the binary

2/2 = 1 0 number from bottom to top.

1/2 = 0 1 The number is 10000 or 0x10

Convert the Part to the Right of the Decimal

This is done by a simple variant of multiplication.
This is easier to show than to describe. Convert 0.375

Number Product Binary

0.375 x 2 = 0.75 0

0.75 x 2 = 1.5 1 Read top to bottom as .011

0.5 x 2 = 1.0 1

Note that the multiplication involves dropping the leading ones from the product terms, so that our products are 0.75, 1.5, 1.0, but we would multiply only the numbers 0.375, 0.75, 0.50, and (of course) 0.0.

Another example: convert 0.71875

Number Product Binary
0.71875 x2 = 1.4375 1
0.4375 x 2 = 0.875 0 Read top to bottom as .10111
0.875 x 2 = 1.75 1 or as .1011100000000 …
0.75 x 2 = 1.5 1 with as many trailing zeroes as you like
0.5 x 2 = 1.0 1
0.0 x 2 = 0.0 0

Convert an “Easy” Example

Consider the decimal number 0.20. What is its binary representation?

Number Product Binary

0.20 · 2 = 0.40 0
0.40 · 2 = 0.80 0

0.80 · 2 = 1.60 1

0.60 · 2 = 1.20 1

0.20 · 2 = 0.40 0

0.40 · 2 = 0.80 0

0.80 · 2 = 1.60 1 but we have seen this – see four lines above.

So 0.20 decimal has binary representation .00 1100 1100 1100 ….

Terminating and Non–Terminating Numbers
A fraction has a terminating representation in base–K notation only if the number can be represented in the form J / (B^K)

Thus the fraction 1/2 has a terminating decimal representation because it is
5 / (10¹). It can also be 50 / (10²), etc.

More on Non–Terminators

What about a decimal representation for 1/3?

If we can generate a terminating decimal representation, there must be positive integers J and K such that 1 / 3 = J / (10^K). But 10 = 2·5, so this becomes

1 / 3 = J / (2^K · 5^K).

Cross multiplying, and recalling that everything is a positive integer, we have

3·J = (2^K · 5^K)

If the equation holds, there must be a “3” on the right hand side. But there cannot be a “3” on this side, as it is only 2’s and 5’s.

Now, 0.20 = 1 / 5 has a terminating binary representation only if it has a representation of the form J / (2^K).

This becomes 1 / 5 = J / (2^K), or 5·J = 2^K. But no 5’s on the RHS.

Because numbers such as 1.60 have no exact binary representation, bankers and others who rely on exact arithmetic prefer BCD arithmetic, in which exact representations are possible.