7.2: Number Bases

Definition:

A number base is the number of digits or combination of digits that a system of counting uses to represent numbers. A base can be any whole number greater than 0. The most commonly used number system is the decimal system, commonly known as base 10. Its popularity as a system of counting is most likely due to the fact that we have 10 fingers.

Example :

The base of any number may be written beside the number. For example, 17₈ is read as 17 base 8, which is 15 in base 10.

Conversion from decimal to other bases

Example : Using positional notation

Binary is the most commonly used non-base 10 system. It is used for coding in computers. Binary is also known as Base 2. This means it is composed of only 0's and 1's. For example 9 in binary/base 2 is 1001. Let's see how this works.

Base 10	Base 2	2^4	2^3	2^2	2^1	2^0
1	1	0	0	0	0	1
9	1001	0	1	0	0	1
16	10000	1	0	0	0	0

Column 2 in the table above represents the binary representation of the decimal number shown in column 1. Columns 3-7 show the expansion of base 2. Note that leading zeros are not normally shown. The Binary system works similarly to the same way base 10 does, only smaller, therefore, requires more digits to make up the same number as in base 10. To illustrate, 16₁₀ = 1(10¹) + 6(10⁰) = 1(2⁴) + 0(2³) + 0(2²) + 0(2¹) + 0(2⁰).

Example :using positional notation

Look at Base 16, also known as the hexadecimal system, another common base when coding and using computer systems. In this case, we use the digits and the letters representing two digits

Base 10	Base 16	16^2	16^1	16^0
100	064	0	6	4
42	02A	0	2	A (10)
124	07C	0	7	C (12)
269	10D	1	0	D (13)

 

Example :Using division by base

Convert into base

Solution

Notice that .

We write the remainders from bottom to top, .

 

Conversion from binary to  octal (base 8) and hexadecimal (base 16)

Example

Convert into base and

Solution

and

Arithmetic in other bases

Example :

Addition in Base 2:

Cheat Table to help with Addition:
0+0=0
0+1=1
1+0=1
1+1=10

Let's try it:

110
+ 101
1011
=====

Check: 110₂ = 6₁₀
101₂ = 5₁₀
1011₂ = 11₁₀

Therefore our addition is correct as 6 + 5 = 11 in base 10.

Let's try another one:
10101₂
+ 11101₂
110010₂
======

Check: 10101₂ = 21
11101₂ = 29
110010₂ = 50

Therefore our addition in correct as 21 + 29 = 50 in base 10.

Example :

Subtraction in Base 2:

Cheat table to help with subtraction:

1-1=0
0-0=0
1-0=1
0-1=1 * This requires a carry 10-1=1. As you will remember, 10 in base 2 is 2.

Let's try it:
10111₂
- 10101₂
00010₂
=====

Check: 10111₂ = 23₁₀
10101₂ = 21₁₀
00010₂ = 2₁₀

Therefore, our subtraction is correct as 23 - 21 = in base 10.

Example :

Find

Therefore,

 

Practical Uses

• Coding
• Time (24 hours) or (60)
• Baking (Dozen)
• Imperial or Metric measurement systems