Number systems Visualizing Decimal and Binary We count

Number systems Visualizing Decimal and Binary

We count in base 10 because people started by counting on their fingers Base 10 is a number system that uses 10 digits: 0 -9 All of our numbers are made with these 10 digits: 532, 699 4, 887, 012 956 14 60, 044, 117, 005

How many fingers do you see? n n n Did you have to count them all? (I hope not) How did you know there were 21? We count how many groups of 10, and then add the single fingers. 2 groups of 10 and 1 extra finger.

Numbers are easy to read because of place value 2, 514 • The ones place has the lowest value and is on the right. • Each place has a value that is 10 times greater than the one to the right of it. • Ones • Tens • Hundreds • Thousands… etc.

To find the value of a number, multiply each digit times its place value. 347 3 hundreds n 4 tens n 7 ones n

To write the number Three thousand Five hundred… n n Put a 3 in the thousands place and a 5 in the hundreds place. Which is correct? a) b) c) d) e) f) 35 350 0035 3050 35000 e) But Why ? ? Because the PLACE where a number appears determines its value.

1 0 0 1 Binary is a number system with only 2 digits 1 0 0 1 1 1 1 Zero and One 1 0 0 1 0 n This works for things like: ¨ Light bulbs or switches that are either on or off ¨ Digital computers that store information using many on/off switches. 1 0 0 1 1 1 1

Binary Numbers are also easy to read because of place value 1011 • The ones place has the lowest value and is on the right. • Each place has a value that is 2 times greater than the one to the right of it. • Ones ( 20) • Twos ( 21) • Fours ( 22) • Eights ( 23) … etc.

16 8 4 2 1 It’s easy to see that each place has twice the number of the one to it’s right. What will be the place value of the number to the left of 16? Hopefully you said 32. It’s fairly easy to double numbers.

To show a number in binary, write a 1 if the card is showing, write a zero if the card is not showing. 001 = 1 010 = 2 011 = 3 100 = 4

0 0 What number is this? 1 0 1 4+1 =5 How is 5 written in binary? There is nothing in the 16’s place There is nothing in the 8’s place There is 1 four There is nothing in the 2’s place There is 1 one SO, the answer is 00101 Or, since extra zeros on the left don’t count 101

0 1 1 Which cards are needed to make the number 12? 8 + 4 = 12 What is the binary number for 12? 01100 Or 1100 0 0

1 1 0 Which cards are needed to make the number 27? 16 + 8 + 2 + 1 = 27 What is the binary number for 27? 11011 1 1

In conclusion: Decimal is our ordinary base 10 number system using the digits 0 -9 n Binary is the base 2 number system using only the digits 0 -1 n All number systems have place value that determines the value of the digit in each place. n

But how do we deal with decimals in double numbers? Think about it… what does the 5 represent in 34. 56 ?

10 -1, 10 -2, 10 -3, etc. 0. 5 = 5/10 n 0. 06 = 6/100 n 0. 001 = 1/1000 n Etc. n

So in binary 0. 1 = ½ n 0. 01 = ¼ n 0. 001 = 1/8 n Etc. n How do you write ¾? n ¨ 0. 11 n How do you write 5/8? ¨ 0. 101

This is why… n It’s nearly impossible to get a perfect fit for some decimal numbers. Come up with the best estimation for: n 1/10 n 1/3 n