# Multiplication and division in different bases a relationship

### Number Systems and Bases – BetterExplained

In a similar manner, we can specify numbers in other "bases" (besides 10), using . Redefine the number to be this smallest positive remainder upon division by. Long division involves multiplication and subtraction. Since those math facts often do not apply in bases other than ten, you have to find ways. The trick to doing base-n addition is to "fake" it by doing decimal arithmetic in your Record the difference as a base-n digit at the bottom of the same column. . There is no carry or borrow or any kind of connection between different columns. . Notice that the result of division of base n numbers is another base n integer.

In these instances, the context of their use usually makes the base clear.

## Numbers in Different Bases

Changing from base 10 to a different base One straight-forward, but inefficient way to convert from base 10 to a different base is to: Determine the higest power of the base that goes into the number a non-zero number of times. Determine how many times this power can be subtracted from the number without the result being negative i. Write this digit down. Redefine the number to be this smallest positive remainder upon division by the power in question Redefine the power to be the power divided by the base.

Go back to step 2, unless the power is now less than one -- in which case, you are done. For example, to convert to base 5, we recall that: This process, however, is inefficient in that one must both know and use the various powers of the desired base.

There is a simpler way!

Consider the remainders seen upon division of the following numbers by 5: In each step above, we are just dividing by 5 and looking at both the quotient and remainder -- no knowledge of higher powers of 5 is necessary! Wonderfully, this technique works in any base. Keep the decimal sum in your head - or write it down on some scratch paper.

Compare the sum in your head with the radix n. A sum less than n is just recorded at the bottom of the same column. Be careful to write the sum as a single base-n digit. Write a 0 as the carry in the next column left.

A sum greater or equal to n requires several steps: Subtract in decimal in your head the radix n from the sum of the digits you already computed. Record the difference as a base-n digit at the bottom of the same column.

### Numbers in Different Bases | The Oxford Math Center

Write a 1 as the carry in the next column left. Any extra carry past the leftmost column is not recorded in the answer. It is called the carry out and is usually stored somewhere else on the computer.

The answer must have the same number of digits as the two operands. Example Adding Base-n Numbers 1.

### Base-n Arithmetic

B 2 C 5 9 second number: Start at the right and add 9 and 2 to get However - write the 11 as a B since that is a base digit. There is a zero carry in the next column over because the 11 is less than the base Second from the right Add the carry 0 and the 5 and the A, but remember that the A is really This is too large greater than or equal to So subtract the base: Also record a carry of 1 in the next column left.

Third from the right Add the carry 1 and the C and the 6, but remember that the C is really Fourth from the right Add the carry 1 and the 2 and the 4.

This is less than 13 and so is recorded into the answer at the bottom.