Ever think that a computer doesn't get what you want it to do? Don't be surprised; it is living in a world where it can only count to 1. This lesson explains number systems, including our base-ten system and a computer's base-two system.
What Is a Number System?
If I were to ask you to write the words for the numbers one through ten, you'd go ahead and write them out. One, two, three, and so on, all the way up through ten. However, if I were to ask you to write them out as numbers, you'd write 1, 2, 3 and then again, all the way up through 10.
One would look like a numeral 1, but the word 'ten' doesn't look like the word 'one.' So why does 10 have a numeral for '1' in the numeral? The answer is simple: you've used all that our number system has to offer for units, so you have to move on to the tens place.
A number system is a system that lets us express numbers in writing. If it weren't for a number system, you'd have to have an infinite number of symbols to express numbers. Luckily, we just have to know ten numerals.
Using Our Base-Ten Number System
We name number systems based on the number of digits that are used in the system. Our standard number system is called base-ten because there are ten base figures. Those are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Any number larger than 9 and we have to move to a different slot.
In the past, you've probably learned about different slots for numerals. The first slot to the left of the decimal is for units. The second is for tens. The third is for hundreds, and so on to infinity. It seems pretty universal, but the truth is that it only works on numbers that are in base-ten. So what would other systems look like? Let's take a look.
Building a Binary System
Computers may be pretty smart, but every chip in a computer is only capable of answering one question: on or off? We express this idea of on or off (or yes or no, if you prefer that question) by 1s and 0s.
A computer can't handle a base-ten number, so the first thing it does is handle it to an idea that it can understand. That happens to be a base-two number, also called a binary number. Whereas base-ten numbers have 10 possible digits, a base-two number only has two: 0 and 1. Let's take a look at how that works. If you were to write zero in base-two, you would simply write 0. Now if you were to write one, you'd just write 1.
But here's where things get tricky. How would you write two? You can't write 2 because there's no symbol for two in base-two. Instead, you have to move a spot to the left and write a 1. This is the twos slot. You would write a zero in the ones spot, just like with the number ten in base-ten, the number is expressed completely in that earlier slot. Whereas slots in base-ten go in units, tens, hundreds, thousands; slots in base-two go ones, twos, fours, eights, sixteens, and so on.
Let's make a bigger number. Let's say that you wanted to write eleven in base-two. First, draw a number of blanks and write below each what place goes over each. Let's say that you draw one for eights, fours, twos, and ones; so four blanks total. Working from left, subtract the value from 11. Eight will go into 11 one time, so write a 1 in the eights place. Then move on the fours. Now you only have three left. Will four go into 3? No, so write a 0. Now the twos. Will two go into 3? Yes, so write a 1. Finally, you have 1 left; as one will go into 1, write a 1 in the ones' spot. Therefore, 1011 is eleven in binary.
Establishing Number Systems
This lesson concept might still be a little fuzzy in your brain, so we're going to build two more different number systems, each with a couple of examples. First, let's look at a base-four system. This is one that would use the numbers 0, 1, 2, and 3. Let's say you wanted to write the number thirty in base-four. What would you do? First, determine your slots. In base-four, that would be fours, sixteens, and sixty-fours. In case you haven't noticed, all you are doing is multiplying the last determined slot by the base system.
Second, starting from the left, work your way over. Will sixty-four go into 30? No, so write a 0. Will sixteen go into 30? Yes, one time, so write a 1. Move on to the fours. How many times will four go into 14? Three, so write a 3. Finally, you are left with 2, so write that in the last spot. That means that 30 in base-four is 0132.
Let's do a base-five system now, but let's convert 206 to base-five. First, what do you think our slots should be? Five, twenty-five, one hundred twenty-five. Remember to put an extra slot on the far right for any leftovers, just like our units spot in base-ten.
Now to the questions. How many times will one hundred and twenty-five go into 206? Once, so put a 1 in that spot. Now that leaves us with 81. How many times will twenty-five go into 81? Three, so write a 3. That leaves us with 6. Five will go in once, meaning we can write a 1, and then we have a 1 as our remainder. Therefore, 206 in base-5 is 1311.
In this lesson, we looked at how a number system works. A number system lets us write with place slots and use the same numbers again instead of having to memorize an infinite amount of numbers.
We tend to use a standard number system called base-ten because there are ten basic figures, but other bases, such as base-two, base-four, and base-five, are important nonetheless. Computers use a base-two system also known as a binary system.
Remember, just as our slots are units, tens, hundreds, and so on, other bases use slots based on their own maximum numbers. Therefore, a binary system is units, twos, fours, eights, and so on, while a base-five is units, fives, twenty-fives, and one hundred twenty-fives.
Number Systems Overview
||a system that lets us express numbers in writing
||standard number system with ten base figures 0 through 10
||a base-two system
Set these goals as you review the lesson above:
- Express knowledge of a number system
- Use the base ten number system
- Build binary and number systems