Express the following number in numers of system with base five,six,seven and eight. (i) 65
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Answer:
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping consecutive binary digits into groups of three (starting from the right). For example, the binary representation for decimal 74 is 1001010.
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Answer:
Way back in the day, we didn’t have base systems! It was uphill both ways, through the snow and blazing heat. When you wanted to count one, you’d write:
l
When you wanted 5, you’d write
lllll
And clearly, 1 + 5 = 6
l + lllll = llllll
This is the simplest way of counting.
Enter The Romans
In Roman numerals, two was one, twice. Three was one, thrice:
one = I
two = II
three = III
However, they decided they could do better than the old tradition of lines in the sand. For five, we could use V to represent lllll and get something like
l + V = Vl
Not bad, eh? And of course, there are many more symbols (L, C, M, etc.) you can use.
The key point is that V and lllll are two ways of encoding the number 5.
Give Each Number A Name
Another breakthrough was realizing that each number can be its own distinct concept. Rather than represent three as a series of ones, give it its own symbol: “3″. Do this from one to nine, and you get the symbols:
1 2 3 4 5 6 7 8 9
The Romans were close, so close, but only gave unique symbols to 5, 10, 50, 100, 1000, etc.
Use Your Position
Now clearly, you can’t give every number its own symbol. There’s simply too many:)
But notice one insight about Roman numerals: they use position of symbols to indicate meaning.
IV means “subtract 1 from 5″
and VI means “add 1 to 5″.
In our number system, we use position in a similar way. We always add and never subtract. And each position is 10 more than the one before it.
So, 35 means “add 3*10 to 5*1″ and 456 means 4*100 + 5*10 + 6*1. This “positional decimal” setup is the Hindu-Arabic number system we use today.
Our Choice Of Base 10
Why did we choose to multiply by 10 each time? Most likely because we have 10 fingers.
One point to realize is you need enough digits to “fill up” until you hit the next number. Let me demonstrate.
If we want to roll the odometer over every 10, so to speak, we need symbols for numbers one through nine; we haven’t reached ten yet. Imagine numbers as ticking slowly upward – at what point do you flip over the next unit and start from nothing?
Enter Zero
And what happens when we reach ten? How do we show we want exactly one “ten” and nothing in the “ones” column?
We use zero, the number that doesn’t exist. Zero is quite a concept, it’s a placeholder, a blank, a space, and a whole lot more. Suffice it to say, Zero is one of the great inventions of all time.
Zero allows us to have an empty placeholder, something the Romans didn’t have. Look how unwieldly their numbers are without it.
George Orwell’s famous novel “1984″ would be “MCMLXXXIV”! Rolls right off the tongue, doesn’t it?:)
Considering Other Bases
Remember that we chose to roll over our odometer every ten. Our counting looks like this:
1
2
3
4
5
6
7
8
9 (uh oh, I’m getting full!)
10 (ticked over – start a new digit)
What if we ticked over at 60 when we counted, like we do for seconds and minutes?
1 second
2
3
4
5
…
58
59
1:00 (60 seconds aka 1 minute. We’ve started a new digit.)
Everything OK so far, right? Note that we use the colon (:) indicate that we are at a new “digit”. In base 10, each digit can stand on its own.
Step-by-step explanation: