NUMBER SYSTEMS

The Decimal (base-10) Number System

Powers of 10
10^-5 = ¹/_100,000 = .00001
10^-4 = ¹/_10,000 = .0001
10^-3 = ¹/_1,000 = .001
10^-2 = ¹/₁₀₀ = .01
10^-1 = ¹/₁₀ = .1
10⁰ = 1
10¹ = 10
10² = 100
10³ = 1,000
10⁴ = 10,000
10⁵ = 100,000

Evaluating a number in base-10

Example: 1,492.76
1,492.76 = (1x1,000) + (4x100) + (9x10) + (2x1) + (7x¹/₁₀) + (6x¹/₁₀₀)
1,492.76 = (1x10³) + (4x10²) + (9x10¹) + (2x10⁰) + (7x10^-1) + (6x10^-2)

From this example, we see that the value of a number is the sum of the products of each digit multiplied by its positional value. That is, the 4 has a positional value of 100 (or 10²), so its value is 400; 9 has a positional value of 10 (or 10¹), so its value is 90, and so. After we get the value of each digit, we add the values: 1,000 + 400 + 90 +2 + .7 + .06 = 1,492.76.

The Binary (base-2) Number System

Powers of 2
2^-5 =¹/₃₂ = .03125
2^-4 =¹/₁₆ = .0625
2^-3 =¹/₈ = .125
2^-2 =¹/₄ = .25
2^-1 =¹/₂ = .5
2⁰ = 1
2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵= 32

Evaluating a number in base-2

Example: 1010.11
1010.11 = (1x8) + (0x4) + (1x2) + (0x1) + (1x¹/₂) + (1x¹/₄)
1010.11 = (1x2³) + (0x2²) +(1x2¹) + (0x2⁰) + (1x2^-1) + (1x2^-2)
In other words, evaluating a number is the same as in base-10, except that it is easier. Since one times any number is that number, and zero times any number is zero, we can rewrite our formula as: 1010.11 = 8+2+¹/₂+¹/₄, which equals 10.75 in base-10.

Decimal to binary conversion

One method of conversion is to use the powers of two, as in the above example. Another method is repeated division (also known as the quotient-and-remainder method). You divide repeatedly by 2, and each time the remainder becomes a digit in the binary number.

Example: 29 in base-10 is equivalent to 11101 in base-2.
Method:
29/2 = 14 with a remainder of 1
14/2 = 7 with a remainder of 0
7/2 = 3 with a remainder of 1
3/2 = 1 with a remainder of 1
1/2 = 0 with a remainder of 1
So, using the remainders in reverse order, the number is 11101.

For a fractional number, the method of conversion is repeated multiplication (also known as the product-and-carry method).
Example: .625 = .101
Method:
.625x2 = .250 with a carry of 1
.250x2 = .500 with a carry of 0
.500x2 = .000 with a carry of 1
So, using the carries, the number is .101.

Applying these methods to a number with both an integer and fractional component:
10.75 = 1010.11 as follows:
10/2 = 5 with a remainder of 0
5/2 = 2 with a remainder of 1
2/2 = 1 with a remainder of 0
1/2 = 0 with a remainder of 1
So, using the remainders in reverse order, the number is 1010
Now for the fractional component:
.75x2 = .50 with a carry of 1
.50x2 = .00 with a carry of 1
So, using the carries, the number is .11.
Putting the two parts together, we get 1010.11

The Hexadecimal (base-16) Number System

Powers of 16
16^-4 = ¹/_65,536 = .000015258789
16^-3 = ¹/_4,096 = .000244140625
16^-2 = ¹/₂₅₆ = .00390625
16^-1 = ¹/₁₆ = .0625
16⁰ = 1
16¹ = 16
16² = 256
16³ = 4,096
16⁴ = 65,536

Evaluating a number in base-16

Example: 3AF.2
3AF.2 = (3x256) + (10x16) + (15x1) + (2x¹/₁₆) = 768 + 160 + 15 + .125 = 943.125
Or we can write the formula as powers of the base:
3AF.2 = (3x16²) + (10x16¹) + (15x16⁰) + (2x16^-1)

Conversions between base-2 and base-16

Even though it is cumbersome to convert to and from base-10, it is easy to go from hex to binary or vice versa.

• Hex to binary: replace each hex digit with 4 bits. Example: 3C7 = 0011 1100 0111
• Binary to hex: divide the number into groups of 4 bits (from the right), and replace each group with a hex digit. Example: 11 1100 0111 = 3C7.

The Octal (base-8) Number System

Powers of 8

8^-4 = ¹/_4,096 = .000244140625
8^-3 = ¹/₅₁₂ = .001953125
8^-2 = ¹/₆₄ = .015625
8^-1 = ¹/₈ = .125
8⁰ = 1
8¹ = 8
8² = 64
8³ = 512
8⁴ = 4,096

Evaluating a number in base-8

Example: 347.2

347.2 = (3x64) + (4x8) + (7x1) + (2x¹/₈) = 192 + 32 + 7 + .25 = 231.25

Or we can write the formula as powers of the base:

347.2 = (3x8²) + (4x8¹) + (7x8⁰) + (2x8^-1)

Binary Arithmetic

Addition

The basic rules are:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 with a carry of 1

Subtraction

Most computer systems perform subtraction by a method known as twos-complement addition. It works like this:

Example: 101 – 11 = 10 (in decimal, 5 – 3 = 2).

Method:

• If the subtrahend (the number to be subtracted) begins with 1, add a leading zero: 011.
• Take the complement of the subtrahend by replacing each digit with its opposite: 100.
• Add 1 to the complement to get the twos-complement: 100 + 1 = 101.
• Now add the twos-complement subtrahend to the minuend (the first number): 101 + 101 = 1010.
• Ignore the final carry and any resulting leading zero: 10.

This may seem like a needlessly involved process, but the reason for it is that the only numeric operation a computer can actually do is addition, so other numeric operations must somehow be reduced to addition.

Multiplication and Division

Multiplication is just repeated addition: "2x3 = ?" is like asking "What do you get when you add three twos?"

Division is just repeated subtraction: "10/3 = ?" is like asking "How many times can you subtract 3 from 10 and what will be left over?"