Operations in That Scary Binary System! Quiz

Answer: 1100

I find the easiest way to add binary numbers is to convert them into base ten and add them together then. You can then change the number back into the binary system. Keep in mind that the last slot in a binary number is 2^0, not 2^1, which means when there are, say, 5 digits in a binary number, the first number is in the 2^4 slot, not the 2^5 slot.

The binary system is based on powers of two, so 101 is equal to (2^2 * 1) + (2^1 * 0) + (2^0 * 1). Simplify that into 4 + 0 + 1, or 5. Do the same with 111, your other binary number. (2^2 * 1) + (2^1 * 1) + (2^0 * 1), which can be simplified to 4 + 2 + 1, which is equal to 7. Take your two numbers that are in base ten, 5 and 7, and add them together to get 12. Change that into a binary number by determining which powers of two add up to twelve. If you try 2^4 (which is 16) first, you will see that 16 is larger than 12, and a 1 cannot be in the 2^4 spot. Try 2^3, which is 8. You will see that it is smaller than 12 and therefore can have a 1 put in its place. Progress on to 2^2, which is 4. You have 4 left over with your 12, so stick a 1 in that place. Place zeros in the 2^1 and 2^0 place value slots, because you have your number. Your binary number for 12 is 1100.

Answer: 39

Convert 11001 by finding the value of each 1. The expression to determine its base ten value is: (2^4 * 1) + (2^3 * 1) + (2^2 * 0) + (2^1 * 0) + (2^0 * 1). This can be further simplified to 16 + 8 + 0 + 0 + 1, which, when added together, equals 25. Add 25 to 14 to get 39.

Answer: 100

First, change both binary numbers into base ten numbers. 1011 = (2^3 * 1) + (2^2 * 0) + (2^1 * 1) + (2^0 * 1) = 8 + 0 + 2 + 1 = 11. The other number is 111 = (2^2 * 1) + (2^1 * 1) + (2^0 * 1) = 4 + 2 + 1 = 7. 11 - 7 = 4. Try converting it into a binary number. 2^2 = 4, so put a 1 in that slot, and zeros in the other to make your binary number: 100.

Answer: 12

First, change 11101 into a base ten number. 11101 = (2^4 * 1) + (2^3 * 1) + (2^2 * 1) + (2^1 * 0) + (2^0 * 1) = 16 + 8 + 4 + 0 + 1, or 29. Subtract 17 from it to get 12.

Answer: 11000

Multiplying in the binary system is a piece of cake. Just multiply the two numbers just like you normally would. 100 x 110 = 11000. If you don't believe me, you can change the numbers into base ten, multiply them, and then change the product back into a binary number: (2^2 * 1) + (2^1 * 0) + (2^0 * 0) = 4 + 0 + 0, or 4. (2^2 * 1) + (2^1 * 1) + (2^0 * 0) = 4 + 2 + 0, or 6. Multiply them together to get 24, and change that back into a binary number. 16 is less than 24, so put a 1 in the 2^4 slot. You have 8 left over, so you can put a 1 in the 2^3 slot and put 0's in the 2^2, 2^1, and 2^0 slots. Your binary number is 11000, just what we got when we multiplied the binary numbers like normal.

Answer: 726

Since the two numbers are in different bases, we can't multiply them. Change 10110 into a base ten number first. 10110 = (2^4 * 1) + (2^3 * 0) + (2^2 * 1) + (2^1 * 1) + (2^0 * 0) = 16 + 0 + 4 + 2 + 0, or 22. Then multiply it by 33 to get 726.

Answer: 11

Once again, we can simply divide the two numbers to get 11. If you don't think this will work, you can go through the time-consuming process of changing the numbers to base ten form, dividing them, and changing the answer back to binary form, which I'll gladly show you how to do. 1100 = (2^3 * 1) + (2^2 * 1) + (2^1 * 0) + (2^0 * 0), which equals 8 + 4 + 0 + 0, or 12.

The other number is 100. (2^2 * 1) + (2^1 * 0) + (2^0 * 0), which equals 4 + 0 + 0, or 4. Divide the two numbers to get 3. Convert it back into a binary number. 2 is less than three, so you can put a one in the 2^1 slot. You have 1 leftover, so go ahead and stick a 1 in the 2^0 slot. 11 is your answer here, too.

It is the same answer as when you divided the two binary numbers.

Answer: 124

First, change 110 into a base ten number. (2^2 * 1) + (2^1 * 1) + (2^0 * 0) = 4 + 2 + 0 = 6. Then, do your division to get the answer 124.

Answer: 1000000

I think the easiest way to solve this problem is to change the numbers into base ten form. 100 = (2^2 * 1) + (2^1 * 0) + (2^0 * 0), which is 4. 11 = (2^1 * 1) + (2^0 * 1) = 2 + 1, or 3. Evaluate 4 to the third power, which would be 4 * 4 * 4, or 64. Change that back into the binary system. If you notice, 64 is a perfect power of 2.

It happens to be the sixth power of 64, so you can stick a one in the 2^6 power slot and stick a zero in the other six slots. There is six slots because the slots are 2^5, 2^4, 2^3, 2^2, 2^1, and 2^0. Your answer is 1000000.

Answer: 4194304

First, change 100 into a base ten number. (2^2 * 1) + (2^1 * 0) + (2^0 * 0), 4 + 0 + 0, or 4. Then evaluate 4 to the 11th power. A calculator would be very helpful. The answer is 4194304.