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Converting in between different number bases is actually relatively simple, yet the reasoning behind it deserve to seem a bit confusing in ~ first. And while the topic of different bases might seem somewhat pointless come you, the climb of computers and also computer graphics has increased the require for expertise of exactly how to occupational with various (non-decimal) basic systems, specifically binary solution (ones and also zeroes) and also hexadecimal solution (the number zero v nine, adhered to by the letters A with F).

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In ours customary base-ten system, we have digits for the number zero through nine. We carry out not have actually a single-digit character for "ten". (The Romans did, in your character "X".) Yes, we create "10", but this represents "1 ten and 0 ones". This is 2 digits; we have actually no single solitary digit that represents "ten".

Instead, once we should count come one an ext than nine, we zero the end the ones shaft and add one to the tens column. As soon as we obtain too big in the tens column -- when we require one much more than nine tens and nine ones ("99"), we zero the end the tens and also ones columns, and add one to the ten-times-ten, or hundreds, column. The next column is the ten-times-ten-times-ten, or thousands, column. And so forth, v each bigger shaft being ten times bigger than the one before. We place digits in each column, informing us how many copies of that strength of ten us need.


The just reason base-ten math seems "natural" and the various other bases don"t is that you"ve been doing base-ten since you were a child. And also (nearly) every civilization has supplied base-ten math most likely for the an easy reason the we have actually ten fingers. If rather we stayed in a cartoon world, wherein we would have actually only 4 fingers on every hand (count them next time you"re watching TV or analysis the comics), climate the "natural" base device would likely have been base-eight, or "octal".

Binary


Let"s look in ~ base-two, or binary, numbers. How would girlfriend write, because that instance, 1210 ("twelve, basic ten") as a binary number? girlfriend would have actually to transform to base-two columns, the analogue of base-ten columns. In basic ten, you have columns or "places" because that 100 = 1, 101 = 10, 102 = 100, 103 = 1000, and also so forth. An in similar way in basic two, you have columns or "places" because that 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, and also so forth.

The first column in base-two math is the systems column. However only "0" or "1" can go in the units column. When you acquire to "two", you uncover that over there is no solitary solitary number that means "two" in base-two math. Instead, you put a "1" in the twos column and a "0" in the units column, describe "1 two and also 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 composed as 112. "Four" is actually two-times-two, so us zero out the twos column and the systems column, and also put a "1" in the fours column; 410 is written in binary type as 1002. Right here is a listing of the first couple of numbers:


swipe come view complete table →

decimal(base 10)

binary(base 2)

expansion

0

0

0 ones

1

1

1 one

2

10

1 two and also zero ones

3

11

1 two and also 1 one

4

100

1 four, 0 twos, and also 0 ones

5

101

1 four, 0 twos, and 1 one

6

110

1 four, 1 two, and also 0 ones

7

111

1 four, 1 two, and 1 one

8

1000

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

9

1001

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

10

1010

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

11

1011

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

12

1100

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

13

1101

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

14

1110

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

15

1111

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

16

10000

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


Converting between binary and decimal number is reasonably simple, as lengthy as friend remember the each number in the binary number to represent a strength of two.

convert 1011001012 come the matching base-ten number.

I will list the number in order, as they appear in the number they"ve provided me. Then, in another row, I"ll counting these digits off from the RIGHT, starting with zero:


The very first row over (labelled "digits") has the number from the binary number; the 2nd row (labelled "numbering") has the power of 2 (the base) equivalent to each digit. Ns will usage this listing to convert each number to the power of two that it 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 numbers to binaries is nearly as simple: just divide by 2.

convert 35710 to the corresponding binary number.

To perform this conversion, I must divide consistently by 2, maintaining track of the remainders as I go. Watch below:


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The over graphic is man on the "live" web page.

As you have the right to see, after separating repeatedly by 2, I ended up through these remainders:


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These remainders call me what the binary number is. I review the number from about the exterior of the division, starting on optimal 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 to 1011001012.


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This an approach of conversion will job-related for converting to any non-decimal base. Simply don"t forget to include that very first digit top top the top, prior to the perform of remainders. If you"re interested, one explanation the why this method works is obtainable here.

See more: The Information About Invoices To Customers Is Tracked By ____ System.

You can convert from base-ten (decimal) to any other base. When you research this topic in class, girlfriend will probably be intended to convert numbers to miscellaneous other bases, for this reason let"s look in ~ a couple of more examples...