Converting in between different number bases is actually fairly simple, however the thinking behind it deserve to seem a bit confusing in ~ first. And also while the object of different bases might seem rather pointless to you, the climb of computers and computer graphics has actually increased the require for expertise of how to work with various (non-decimal) base systems, particularly binary equipment (ones and zeroes) and hexadecimal solution (the numbers zero with nine, complied with by the letters A with F).

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In ours customary base-ten system, we have digits for the number zero with nine. We perform not have actually a single-digit numeral for "ten". (The Romans did, in their character "X".) Yes, we compose "10", however this stands for "1 ten and also 0 ones". This is 2 digits; we have actually no solitary solitary digit that stands for "ten".

Instead, once we must count to one more than nine, us zero the end the ones shaft and add one to the tens column. As soon as we gain too huge in the tens obelisk -- when we require one more than nine tens and nine ones ("99"), us zero out the tens and ones columns, and include one to the ten-times-ten, or hundreds, column. The following column is the ten-times-ten-times-ten, or thousands, column. And also so forth, with each bigger column being ten times bigger than the one before. We place digits in every column, telling us how many copies of that strength of ten us need.

The just reason base-ten math appears "natural" and the various other bases don"t is the you"ve been doing base-ten since you were a child. And (nearly) every world has used base-ten math more than likely for the basic reason that we have ten fingers. If rather we stayed in a cartoon world, wherein we would have actually only 4 fingers on every hand (count them following time you"re city hall TV or analysis the comics), climate the "natural" base system would likely have been base-eight, or "octal".


Let"s look at base-two, or binary, numbers. Just how would girlfriend write, because that instance, 1210 ("twelve, base ten") together a binary number? girlfriend would have to convert to base-two columns, the analogue that base-ten columns. In basic ten, you have actually columns or "places" because that 100 = 1, 101 = 10, 102 = 100, 103 = 1000, and also so forth. Similarly in basic two, you have actually columns or "places" for 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, and also so forth.

The very first column in base-two mathematics is the units column. However only "0" or "1" have the right to go in the devices column. When you gain to "two", you uncover that over there is no single solitary number that stands for "two" in base-two math. Instead, you put a "1" in the twos column and a "0" in the devices column, denote "1 two and 0 ones". The base-ten "two" (210) is written in binary as 102.

A "three" in base 2 is in reality "1 two and 1 one", so the is created as 112. "Four" is in reality two-times-two, so we zero out the twos column and also the devices column, and put a "1" in the fours column; 410 is composed in binary form as 1002. Here is a listing the the first couple of numbers:

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decimal(base 10)

binary(base 2)




0 ones



1 one



1 two and also zero ones



1 two and also 1 one



1 four, 0 twos, and 0 ones



1 four, 0 twos, and also 1 one



1 four, 1 two, and 0 ones



1 four, 1 two, and also 1 one



1 eight, 0 fours, 0 twos, and also 0 ones



1 eight, 0 fours, 0 twos, and 1 ones



1 eight, 0 fours, 1 two, and 0 ones



1 eight, 0 fours, 1 two, and also 1 one



1 eight, 1 four, 0 twos, and also 0 ones



1 eight, 1 four, 0 twos, and 1 one



1 eight, 1 four, 1 two, and 0 ones



1 eight, 1 four, 1 two, and also 1 one



1 sixteen, 0 eights, 0 fours, 0 twos, and 0 ones

Converting in between binary and also decimal numbers is reasonably simple, as long as friend remember that each digit in the binary number to represent a strength of two.

transform 1011001012 to the matching base-ten number.

I will list the number in order, together they appear in the number they"ve offered me. Then, in an additional row, I"ll counting these digits off from the RIGHT, beginning with zero:

The first row above (labelled "digits") consists of the digits from the binary number; the 2nd row (labelled "numbering") includes the power of 2 (the base) corresponding to each digit. Ns will use this listing to convert each digit to the strength of two that that represents:

1×28 + 0×27 + 1×26 + 1×25 + 0×24 + 0×23 + 1×22 + 0×21 + 1×20

= 1×256 + 0×128 + 1×64 + 1×32 + 0×16 + 0×8 + 1×4 + 0×2 + 1×1

= 256 + 64 + 32 + 4 + 1

= 357

Converting decimal number to binaries is almost as simple: simply divide by 2.

convert 35710 come the corresponding binary number.

To carry out this conversion, I have to divide repetitively by 2, maintaining track of the remainders as I go. Clock below:


The above graphic is man on the "live" net page.

As you can see, after splitting repeatedly by 2, I ended up through these remainders:


These remainders call me what the binary number is. I review the numbers from about the outside of the division, starting on top with the last value and its remainder, and wrapping my means around and also down the right-hand next of the sequential division. Then:

35710 converts come 1011001012.




This technique of conversion will occupational for converting to any kind of non-decimal base. Simply don"t forget to encompass that an initial digit on the top, prior to the perform of remainders. If you"re interested, an explanation that why this an approach works is easily accessible here.

See more: Convert 180 Ml Equals How Many Ounces ? What Is 180 Milliliters In Fluid Ounces

You can convert from base-ten (decimal) to any kind of other base. Once you study this subject in class, friend will more than likely be expected to transform numbers to various other bases, so let"s look at a few more examples...