What is the base 5 representation of the number 465∨8
2 answers:
<h3>Base 8 to Base 10: Avoiding some headaches</h3>
To avoid the hassle of having to learn the base 8 multiplication table, let's convert the number to our more familiar decimal system. The same way each digit in the decimal number 248 represents a multiple of a power of 10, each digit in
To find the base 5 representation of number, we need to "regroup" these places in terms of powers of 5. That means that every group of 5 1s we can group into 1 5, every group of 5 5s we can group into 1 25, every group of 5 25s we can group into 1 125, and so on. All of the "leftovers" are what give us the digits of our number.
To start converting 309 to base 5, let's figure out how many groups of 5 we have, and how many 1s we have left over.
309 / 5 = 61 R 4, so we can make <em>61 groups of 5</em> with <em>4 1s left over</em>. We're going to put that 4 in the one's place. Now, bundle those 61 groups of 5 in to groups of 5:
61 / 5 = 12 R 1, so we can make <em>12 groups of 5 5s (or 12 groups of 25)</em> with <em>1 5 left over</em>. We'll put a 1 in the 5's place and group the 25s together now.
12 / 5 = 2 R 2. That's <em>2 groups of 5 25s (or 2 groups of 125), </em>and we've got <em>2 25s left over</em>. That 2's gonna go in the 25's place.
2 / 5 = 0 R 2. Well, we weren't able to bundle those 125s into any larger groups, so all we have left is to put those 2 leftover 125s in the 125s place! We're done with the hard part.
<h3>Writing our number out</h3>Now that we've handled the regrouping, let's use those leftovers to put our number together. Let's lay out the digits we have in each of our places:
- 1s place: 4
- 5s place: 1
- 25s place: 2
- 125s place: 2
It's conventional to write numbers in any representation left-to-right from <em>highest</em> place value to <em>lowest</em>, so starting at the 125s place, our base 5 representation will be
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