[SOLVED] Addition of Number Bases (…in 3 steps)

Hello,

Welcome to the simplest lesson on addition of number bases. How to add in any base with solved examples in base 2 and 5

At the end of this lesson, you will be able to:

• Count in different number bases.
• Add numbers in any base.

How to add in any base

Take note.

Counting is generally done in base 10, i.e. dinary or decimal.

And the highest digit of a base is always ONE LESS than the base.

E.g.

• base ten: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• base seven: 0, 1, 2, 3, 4, 5, 6.
• base five: 0, 1, 2, 3, 4.
• base two: 0, 1.

#1. Addition of number base two with example

Add 110111₂ to 10100₂

\begin{matrix}
~~~1~1~0~1~1~1 \\
+~~~1~0~1~0~0 \\ \hline
1~0~0~1~0~1~1\\ \hline
\end{matrix}

EXPLANATION

Starting from the back like in your normal addition…

• 1 + 0 = 1
• 1 + 0 = 1
• 1 + 1 = 2

When a number gotten from the addition of any base is greater than or equal to the base, divide that number by the base and take the remainder from bottom to top.

• (2) is equal to the base two.
• So, you divide your answer (2) by the base (2).
• Then count the remainder from bottom to top.

\def\arraystretch{1.5}
   \begin{array}{c:c:c}
   2 & 2 & r \\ \hline
   2 & 1 & 0 \\
   \hdashline
   & 0 & 1
\end{array}\uparrow

• So (1 + 1) in base two = 1, 0.
• Write the last digit (0).
• Carry the remaining digit (1) over to the next addition.
• 0 + 0 + (1) = 1
• 1 + 1 = (1, 0)
• Write the last digit (0)
• Carry (1) over to the final addition.
• 1 + (1) = 1,0
• Write down all the digits since there are NO more numbers to add in the question.

\therefore 110111+10100
=1001011

#2. Example of addition in base five

Add in base five: 44₅ + 42₅

\begin{matrix}
~~~4~4 \\
+4~2 \\ \hline
1~4~1\\ \hline
\end{matrix}

EXPLANATION

Start from behind like in your normal addition and remember the calculation is in base five.

• (4 + 2) = 6
• (6) is greater than the base five, so you divide the (6) by the base (5).

\def\arraystretch{1.5}
   \begin{array}{c:c:c}
   5 & 6 & r \\ \hline
   5 & 1 & 1 \\
   \hdashline
   & 0 & 1
\end{array}\uparrow

• So, (4 + 2) in base five = 1, 1.
• Write the last digit (1).
• Carry the other (1) over to the next addition.
• (1) + 4 + 4 = 9
• (9) is greater than our base five, so you divide it by the base.

\def\arraystretch{1.5}
   \begin{array}{c:c:c}
   5 & 9 & r \\ \hline
   5 & 1 & 4 \\
   \hdashline
   & 0 & 1
\end{array}\uparrow

• So, (9) = 1, 4 in base five.
• Since there is NO other number to add in the question, write down your 1 and 4.

\therefore 44+42~(...in~base~five)=141

Practice questions on addition of number bases

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