University:
Boston University
Course:
CGS MA 121 | Calculus
Academic year:

2021
Views:

453

Pages:

16
Author:

Konner Hansen

Converting Base-10 to Binary! • Let’s convert the number 25 to binary! • First we need to find the largest binary digit that has a value less than 25. – In this case it is 24: 16 – 25 wouldn’t work because it is 32, which is bigger than 25. • So now we know that the largest binary digit for this number will be the 16’s place. Converting Base-10 to Binary! ____ ____ ____ ____ ____ 16’s place 8’s place 4’s place 2’s place 1’s place 24 23 22 21 20 • We chose the 16’s place to be our first digit because 16 is the largest number that can fit inside 25. • So we put a 1 in the 16’s place, indicating that 16 is part of our number. Converting Base-10 to Binary! ____ ____ ____ ____ ____ 16’s place 8’s place 4’s place 2’s place 1’s place • So, now 16 out of our total 25 is accounted for. Let’s take care of the remainder. • 25 – 16 = 9 • Now we go to the next digit, the 8’s place. Does an 8 fit inside 9 – our remainder? • Yes it does!! Converting Base-10 to Binary! ____ ____ ____ ____ ____ 16’s place 8’s place 4’s place 2’s place 1’s place • Now 8 out of the remainder 9 is taken care of. • 9–8=1 • Does the next digit – the 4’s place – fit inside this remainder? • Nope! So, we have to put a 0 at the 4’s place because 1 is smaller than 4. Converting Base-10 to Binary! ____ ____ ____ ____ ____ 16’s place 8’s place 4’s place 2’s place 1’s place • Let’s see if the next digit can take care of our remainder (which is still 1). • The next digit is the 2’s place. Can this digit fit inside our remainder? • No, it can’t either, because 1 is smaller than 2. We have to put a 0 here too. Converting Base-10 to Binary! ____ ____ ____ ____ ____ 16’s place 8’s place 4’s place 2’s place 1’s place • One last try to get rid of our remainder (still 1)! • The last thing we have is the 1’s place. Can a 1 fit inside our remainder? • Yes! 1 is equal to 1! • We have no remainder left now, because 1 – 1 = 0! • We’re done! Converting Binary into Base-10! ____ ____ ____ ____ ____ 16’s place 8’s place 4’s place 2’s place 1’s place • Now let’s convert our number back! • All we have to do is take each binary digit, and figure out how much it is worth in base-10. • 0 means that the digit doesn’t add anything to our number • 1 means it adds the value of the place it’s in • This way is easier! Converting Binary into Base-10! ____ ____ ____ ____ ____ 16’s place 8’s place 4’s place 2’s place 1’s place • • • • • 1 in the 16’s place. Add 16. 1 in the 8’s place. Add 8. 0 in the 4’s place. Nothing added 0 in the 2’s place. Nothing added 1 in the 1’s place. Add 1. 16 8 0 0 +1 TOTAL: Practice! 1. Is the following number written in binary form? 121011 – No! Binary only has 1s and 0s. Practice! 2. What is this binary number in the base-10 system? 111 – It is (4 x 1) + (2 x 1) + (1 x 1), which is 7! Practice! 3. What is this base-10 number in binary? 14 – 14 has 1 eight, 1 four, 1 two, and 0 ones. So it’s 1110! Practice! 3. What is this base-10 number in binary? 11 – 11 has 1 eight, 0 fours, 1 two, and 1 one. So it’s 1011! Why does binary matter? A computer has many switches inside it that tell it what to do. The computer will do different things, depending on which ones are switched ON and which are OFF. To a computer, an ON switch is represented by 1 and an OFF switch is represented by 0. Why does anyone use binary? Because computers only understand things in terms of ON and OFF, a system with only two options for digit values makes a lot sense (OFF = 0, ON = 1). This makes it very easy for computers to express everything happening inside them as a bunch of 0s and 1s. Why does anyone use binary? There is a particular assortment of ON and OFF switches for everything you do on a computer. ANYTHING you do on computer can be represented as a very long string of binary. It sounds crazy until you realize there are a huge amount of switches in your computer – and SO many ways the whole system can be arranged. Actually, it still sounds pretty amazing!

What Is Binary Lecture Note Part 2

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