Multiplication of Number Bases

Hello,

Welcome to the simplest lesson on multiplication of number bases. How to multiply 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.
• Multiply numbers in any base.

How to multiply 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: Multiplication of numbers in base two

Multiply 110111₂ × 10100₂

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

EXPLANATION (STEP 1: MULTIPLY)

Start from behind like in your normal multiplication…

First row

• 0 × 1 = 0
• 0 × 1 = 0
• 0 × 1 = 0
• 0 × 0 = 0
• 0 × 1 = 0
• 0 × 1 = 0

Second row is the same with first row (000000) because it is the next zero that is multiplying the same digits above.

Third row.

• 1 × 1 = 1
• 1 × 1 = 1
• 1 × 1 = 1
• 1 × 0 = 0
• 1 × 1 = 1
• 1 × 1 = 1

Fourth row is the same with first and second row (000000) because it is another zero that is multiplying the digits above.

Fifth row is the same third row (110111) because it is (1) that is multiplying the digits above.

STEP 2: ADD

• 0 + nothing = 0
• 0 + 0 = 0
• 0 + 0 + 1 = 1
• 0 + 0 + 1 + 0 = 0
• 0 + 0 + 1 + 0 + 1 = 2

Remember that 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.

See How To do this in our LESSON ON ADDITION OF NUMBER BASES.

The answer (2) is equal to the base we are calculating in, so we do the above division to get the remainder (1, 0).

Write the (0), carry (1) over to the next addition.

• 0 + 0 + 0 + 0 + 1 (+1) = 2

(2) is equal to our base number, so we divide to get the remainder (1, 0).

Write the (0) and carry (1) again over to the next addition.

• 0 + 1 + 0 + 1 (+1) = 3

(3) is bigger than our base number, so we divide and get the remainder (1, 1).

Write the last (1) and carry the other (1) to the next addition.

• 1 + 0 + 0 (+1) = 2

Divide again and carry over (1).

0 + 1 (+1) = 2

Divide again.

• 1 (+1) = 2

Write out the (1, 0) since there are no more additions to carry over to.