[SOLVED] Subtraction of Number Bases (…in 2 steps)

Hello,

Welcome to the simplest lesson on subtraction of number bases. How to subtract in any base with solved examples in base 2, 5, 8 and questions for you to practice.

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

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

How to subtract 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. Subtraction of number base two with example

Subtract 110₂ from 11001₂

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

EXPLANATION

Start from behind like in your normal subtraction…

• 1 – 0 = 1
• 0 – 1 = impossible (so we borrow)

When you borrow in number base subtraction, the (1) you borrowed becomes the same number with the base you are calculating in.

• So, (0 – 1) becomes (2 – 1) since we borrowed and we’re calculating in BASE TWO.

Remember we borrowed from (1) to the first zero and from that first zero to the second zero. So the first zero which was (2) when it borrowed is now (1) after it gave out.

• And (1 – 1) = 0.

Since we borrowed from that (1), there is nothing left of it.

• (0 – nothing) = 0.
• (1 – nothing) = 1.

\therefore 11001 – 110=10011

#2. Example of subtraction in base five

Subtract in base five: 404₅ – 323₅

\begin{matrix}
~~~4~0~4 \\
-3~2~3 \\ \hline
~~~0~3~1\\ \hline
\end{matrix}

EXPLANATION

Start from behind like your normal subtraction and also remember the calculation is in BASE FIVE.

• (4 – 3) = 1
• (0 – 2) is impossible, so we borrow.

Again, when you borrow in number base subtraction, the (1) you borrowed becomes the same number with the base you are calculating in.

• So, (0 – 2) becomes (5 – 2) since we borrowed and we’re calculating in BASE FIVE.

Remember we borrowed from (4) so it is now (3).

• And (3 – 3) = 0

\therefore 404-323~(...base~five)=141

#3. Subtraction of numbers in base eight

Subtract in base eight: 7243₈ – 4536₈

\begin{matrix}
~~~7~2~4~3 \\
-4~5~3~6 \\ \hline
~~~2~5~0~5\\ \hline
\end{matrix}

EXPLANATION

Start from behind like your normal subtraction and also remember the calculation is in BASE EIGHT.

• (3 – 6) is impossible (so we borrow).

Again, when you borrow in number base subtraction, the (1) you borrowed becomes the same number with the base you are calculating in.

• So, (3 – 6) becomes (11 – 6) since we added what we borrowed to 3 and calculating in BASE EIGHT.

Remember we borrowed from (4) so it is now (3).

• And (3 – 3) = 0
• (2 – 5) is impossible, so we borrow.

Always remember that when you borrow in number base subtraction, the (1) you borrowed becomes the same number with the base you are calculating in.

• So, (2 – 5) becomes (10 – 5) after adding what we borrowed to 2 while still calculating in BASE EIGHT.

Remember we borrowed from (7) so it is remaining (6).

• And (6 – 4) = 2

\therefore 7243-4536~(...base~eight)=2505

Practice questions on addition of number bases

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