Introduction To Base Two

 

Introduction to Number Bases (Base Two)

In our everyday lives, we use Base Ten (Decimal), which relies on ten digits (0 through 9). However, computers and digital systems don't understand ten digits—they operate on electricity, which is either on or off.

This is where Base Two (Binary) comes in. The binary system uses only two digits: 0 and 1.

• 0 represents "off" (low voltage).
• 1 represents "on" (high voltage).

How Base Two Works

Just like base ten uses powers of 10 (1, 10, 100, 1000), base two uses powers of 2. The place values double each time you move to the left:

24 (Sixteens)	23 (Eights)	22 (Fours)	21 (Twos)	20 (Ones)
16	8	4	2	1

To show a number is in a specific base, we write a small subscript. For example, 13₁₀ (thirteen in base ten) is written as 1101₂ in base two.

Converting Base Ten to Base Two

To convert a number from base ten to base two, we use the repeated division method. Divide the number by 2, write down the remainder, and repeat until you get 0. Then, read the remainders from the bottom to the top.

Question 1: Convert 6₁₀ to Base Two

Solution:
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Answer: 6₁₀ = 110₂

Question 2: Convert 9₁₀ to Base Two

Solution:
• 9 ÷ 2 = 4 remainder 1
• 4 ÷ 2 = 2 remainder 0
• 2 ÷ 2 = 1 remainder 0
• 1 ÷ 2 = 0 remainder 1

Answer: 9₁₀ = 1001₂

Question 3: Convert 12₁₀ to Base Two

Solution:
• 12 ÷ 2 = 6 remainder 0
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Answer: 12₁₀ = 1100₂

Question 4: Convert 15₁₀ to Base Two

Solution:
• 15 ÷ 2 = 7 remainder 1
• 7 ÷ 2 = 3 remainder 1
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Answer: 15₁₀ = 1111₂

Question 5: Convert 20₁₀ to Base Two

Solution:
• 20 ÷ 2 = 10 remainder 0
• 10 ÷ 2 = 5 remainder 0
• 5 ÷ 2 = 2 remainder 1
• 2 ÷ 2 = 1 remainder 0
• 1 ÷ 2 = 0 remainder 1

Answer: 20₁₀ = 10100₂