Question

complete the following operations using the standard algorithm then order the answer from smallest to largest.

Answer 1

9514 1404 393

Answer:

• 1000100₂
• 10100₄
• 643₈

Explanation:

`\begin{array}{rrr}100111_2&732_8&3213_4\\+11101_2&-67_8&+221_4\\\cline{1-3}1000100_2&643_8&10100_4\end{array}`

It can be convenient to convert these to base 8 for comparison. (Bases 2, 4, and 8 are easily interconverted by regrouping the base-2 bits of the number.) Using periods to mark groups of bits, we have ...

1000100₂ ⇒ 1.000.100₂ ⇒ 104₈

10100₄ ⇒ 01.00.01.00.00₂ ⇒ 0.100.010.000₂ ⇒ 0420₈

Then the numbers in order are, smallest to largest, ...

• 1000100₂ ⇒ 104₈
• 10100₄ ⇒ 420₈
• 643₈

_____

When the sum of digits in a column equals or exceeds the base, then 1 is carried to the next column to the left, and the excess is written as the sum of the column. In base 2, this means 1+1 = 10. In base 4, this means 3+1=10, or 1+2+2=11.

Subtraction of a larger digit from a smaller one can be done by "borrowing" 1 (the amount of the base) from the next column to the left. In base 8, for example, this means 32-7 = 23.

These rules are identical to the rules that apply in base ten arithmetic.