CHAPTER 5 SEC 3

Calculating in Other Bases

Bases

The Babylonian had a base of 60.

The Mayan system used a base of 20.

The Hindu-Arabic we use today is in base 10.

What other bases do we use everyday ?

Base 2 (the binary)The use of computers and cell phone.The only numbers that are used are zeros and ones.

Other bases

Base 21, 102, 112, 1002, 1012, 1102, …

Base 51, 2, 3, 4, 105, 115, 125, 135, 145, 205, …

How about base 7?

Each number has a place holder, for example given the number 4859 . The subscript 9 is the base that we will have to work in. The 5 is the ones, the 8 is the tens, and the 4 is in the hundreds.

What we want to do is to expand,4x92 + 8x91 + 5x90

= 4x81 +72+5 = 324+77 = 401

What we are going to be trying to do is to convert a different base into base 10.

For example, covert 24, 4567 into base-10.

Hindu-Arabic into a different base.

Write 768 into base 5.

Now we read the remainders in reverse order. Therefore our number is 10,3445.

724/5 = 144r4

144/5 = 28 r 4

28/5 = 5 r 3

5/5 = 1 r 0

1/5 = 1

Addition

Base 6 Addition

+ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 106 2 2 3 4 5 106 116 3 3 4 5 106 116 126 4 4 5 106 116 126 136 5 5 106 116 126 136 146

Try these problems

5536 + 2456

12426

13256 – 4536

4326

Multiplication

Multiplication base 6

x 0 1 2 3 4 5 0 0 0 0 0 0 0 1 0 1 2 3 4 5 2 0 2 4 106 126 146 3 0 3 106 136 206 236 4 0 4 126 206 246 326 5 0 5 146 236 326 416