in this video we take a look at the different number systems or bases that you need to know about these include binary dna and hexadecimal [Music] so deanery or the base 10 number system is the number system which you will be most familiar with it's the one that you've been learning ever since you're a small child ever since you first started to count it contains just 10 unique digits 0 through to 9. hence why it's known as base 10 because there's 10 unique digits available now incidentally why do you think our number system has evolved in this way well the answer is that no one actually really knows the answer but it could quite possibly have something to do with the fact that we have a total of 10 fingers and thumbs so it became natural to count up to 10 on our digits with the deanery system we have no unique digit for the number 10 or indeed for any number higher than 9 we have to put the digit 1 and 0 together in order to get the number 10. so let's look at how that actually works with larger numbers well we'll start with the 10 and obviously i've written that out here now i've got two leading zeros on the beginning but that doesn't affect the fact that this is the number 10 0 0 1 0. you will know from basic maths at primary school that the weighting of the headings in each of these columns goes up by 10 each time so what i have here is naught of thousands plus nor to hundreds plus one in the tens column plus not in the ones column so the number ten if we take a bigger number now so four two seven three the exact same principle is applied the column on the far left is our thousands columns so we have four times a thousand we add that to the next column which is two lots of a hundred we add that to seven lots of ten and three lots of ones with four thousand two hundred and seventy three now of course we don't perform that calculation ahead when we see the number four two seven three we are so used to it now that we simply read it straight out as 4273 notice once again how the column weightings that's the headings are increasing by a factor of 10 every time we move one space to the left and this is because decimal is a base 10 number system now along with our base 10 dna number system that we're all well aware of there are two other base number systems you have to become familiar with for gcse and we're going to take a look at those now and that's base 2 binary and base 16 hexadecimal so with the base 2 binary number system we only have two unique digits so a zero and a one and that's it all other numbers in binary must be made up of a combination of these two digits so what is the number i've got represented here well the first thing you'll notice is the weighting of the column headings has changed starting on the right we have the ones column then the twos then the fours then the eighths it's doubling each time or timesing by 2 and that's because we have a base 2 number system so we just apply the same rules we did for the decimal system here i have naught lots of eights added to naught lots of fours i've got one two and i'm adding that to one one so the number one one in binary is three in decimal and you need to read this as 1 1 and obviously not 11 because an 11 is a decimal number so let's do exactly the same thing but this time for a slightly larger number the process although is identical starting on the left here i have a one in the eight column so i've got one lots of eight i'm adding that to naught lots of four i'm adding that to one lot of two and finally one lot of one so i have an eight plus a two plus a one so this number is eleven in decimal so the binary number one zero one one is eleven in decimal and again just to reiterate the weighting of the column headings is times in by 2 every time we move to the left because binary is a base 2 number system to represent bigger numbers you obviously need to add more columns in the exam you need to be able to read write and understand numbers of up to 16 bits in size so that's 16 columns this obviously gives us a range of numbers that you'll need to be able to represent starting on the right with 16 bits we have a column weighting of 1 and then doubling by 2 our column weighting goes up to 2 4 8 16 all the way through to the most significant column representing the value 32 768. when considering positive numbers this means if we put a naught in every column the smallest number that can be represented in 16 bits is zero and if we put a one in every column the largest number that can be represented is sixty five thousand five hundred and thirty five so hexadecimal is a base sixteen number system and it follows exactly the same principles as the other number systems we've just been looking at the only difference is with hex we have 16 unique digits now this obviously presents us with a bit of a unique problem what do we use to represent the hex digits 10 to 15 we can't simply use our decimal numbers 1 0 for 10 or 1 5 for 15 as these are two digits stuck together well we simply choose to replace digits 10 to 15 with the alphabetic letters a through f so in hex we have 16 unique digits representing naught to 15 naught 1 2 3 4 5 6 7 8 and 9 and then a representing 10 from decimal through to 15 for f so let's just summarize and recap those three base number systems and look at them all side by side counting up from zero so in the left column we have base 10 dna followed by base 16 hex decimal followed by base 2 binary so all those number systems can represent the number zero in a single digit and they can all represent one with a single digit of course as soon as we get to 2 we can represent a 2 in base 10 that's fine and also in hex but in binary we've now run out of unique digits we only have a 0 and a 1 available we now have to combine those zeros and ones as shown earlier in order to represent the dna value 10. so in binary the dna value 10 is 1 0 remember it's important to read that as 1 0 and not the number 10. we can proceed in a likewise fashion all the way up until we reach the dna value 9. now of course after that we don't have a single digit in the dna system anymore for representing the digit 10 so we have to combine digits and again in hex we now have to do something special as described earlier and we have to switch to using letters because hex allows us to represent values above 10 in dna in a single digit so in dna we have one zero or ten in hex we have a and in binary we have one zero one zero this continues all the way up to the dna value 15 which is the hex equivalent f and the binary equivalent 1 1 1 1. of course we can carry on going above that and as soon as we do hex no longer has a single digit which can represent a value so we'd now have to start combining values just like we have been in dna and binary so let's just quickly recap computers use different base number systems there's binary which is base two which contains just a zero or a one it's easy to represent two states in binary such as on off or a high or low voltage dna which is the number system we're most familiar with is base 10 and contains unique digits naught through nine and finally hexadecimal base 16 which contains the digits naught through nine followed by a through f hex numbers can be expressed more compactly than binary numbers and this has several advantages which we take a look at in another video [Music] you