Converting between different number bases is actually fairly simple, but the reasoning behind it can seem a little bit confmaking use of at first. And while the topic of various bases might seem somewhat pointless to you, the rise of computers and also computer system graphics has boosted the need for knowledge of exactly how to occupational through various (non-decimal) base systems, especially binary units (ones and also zeroes) and hexadecimal systems (the numbers zero through nine, complied with by the letters A via F).

You are watching: 10 to the -2 In our customary base-ten system, we have actually digits for the numbers zero through nine. We carry out not have actually a single-digit numeral for "ten". (The Romans did, in their character "X".) Yes, we write "10", but this stands for "1 ten and also 0 ones". This is two digits; we have no single solitary digit that means "ten".

Instead, when we should count to an additional than nine, we zero out the ones column and add one to the tens column. When we get also big in the tens column -- as soon as we require one more than nine 10s and also nine ones ("99"), we zero out the tens and ones columns, and include 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, with each bigger column being ten times bigger than the one before. We area digits in each column, telling us just how many kind of duplicates of that power of ten we need.

The just factor base-ten math seems "natural" and also the other bases don"t is that you"ve been doing base-ten considering that you were a kid. And (nearly) every civilization has offered base-ten math probably for the easy factor that we have actually ten fingers. If rather we stayed in a cartoon human being, wbelow we would certainly have actually only 4 fingers on each hand also (count them following time you"re watching TV or reading the comics), then the "natural" base system would certainly likely have been base-eight, or "octal".

## Binary

Let"s look at base-two, or binary, numbers. How would you write, for circumstances, 1210 ("twelve, base ten") as a binary number? You would certainly need to transform to base-2 columns, the analogue of base-ten columns. In base ten, you have columns or "places" for 100 = 1, 101 = 10, 102 = 100, 103 = 1000, and so forth. Similarly in base 2, you have actually columns or "places" for 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, and also so forth.

The first column in base-two math is the devices column. But only "0" or "1" have the right to go in the devices column. When you obtain to "two", you find that there is no single solitary digit that means "two" in base-2 math. Instead, you put a "1" in the twos column and also a "0" in the units column, indicating "1 2 and 0 ones". The base-ten "two" (210) is composed in binary as 102.

A "three" in base two is actually "1 2 and 1 one", so it is written as 112. "Four" is actually two-times-two, so we zero out the twos column and also the systems column, and also put a "1" in the fours column; 410 is written in binary develop as 1002. Here is a listing of the first few numbers:

swipe to check out full table →

 decimal(base 10) binary(base 2) expansion 0 0 0 ones 1 1 1 one 2 10 1 two and zero ones 3 11 1 2 and 1 one 4 100 1 4, 0 twos, and 0 ones 5 101 1 four, 0 twos, and 1 one 6 110 1 four, 1 two, and also 0 ones 7 111 1 4, 1 two, and also 1 one 8 1000 1 eight, 0 fours, 0 twos, and 0 ones 9 1001 1 eight, 0 fours, 0 twos, and also 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 4, 0 twos, and also 1 one 14 1110 1 eight, 1 4, 1 2, and 0 ones 15 1111 1 eight, 1 four, 1 2, and 1 one 16 10000 1 sixteen, 0 eights, 0 fours, 0 twos, and also 0 ones

Converting between binary and also decimal numbers is sensibly straightforward, as long as you remember that each digit in the binary number represents a power of 2.

Convert 1011001012 to the equivalent base-ten number.

I will list the digits in order, as they appear in the number they"ve given me. Then, in an additional row, I"ll count these digits off from the RIGHT, founding through zero:

The first row over (labelled "digits") has the digits from the binary number; the second row (labelled "numbering") consists of the power of 2 (the base) matching to each digit. I will usage this listing to convert each digit to the power of 2 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 matching binary number.

To perform this convariation, I have to divide repetitively by 2, keeping track of the remainders as I go. Watch below: The above graphic is animated on the "live" internet web page.

As you have the right to watch, after separating consistently by 2, I ended up through these remainders: These remainders tell me what the binary number is. I read the numbers from around the external of the division, founding on peak through the final worth and also its remainder, and wrapping my way roughly and down the right-hand also side of the sequential department. Then:

35710 converts to 1011001012.

Affiliate

Affiliate This method of convariation will certainly work for converting to any kind of non-decimal base. Just don"t forobtain to incorporate that first digit on the peak, prior to the list of remainders. If you"re interested, an explanation of why this strategy works is obtainable right here.

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

You have the right to convert from base-ten (decimal) to any various other base. When you examine this topic in course, you will certainly probably be meant to convert numbers to various various other bases, so let"s look at a few more examples...