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# Number Systems: Base Ten

- We use the
**decimal** number system
- This number system uses
**base** 10
- Another word for base is
**radix**
- And the
**digits** 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

- This is a
** positional** number system
- The position of the digit in a number has meaning
- IE the number 42 and the number 24 are different.

- In this system
- The leftmost digit is called the Most Significant Digit (MSD)
- The rightmost digit is called the Least Significant Digit (LSD)
- In the number 7356
- 7 is the most significant digit
- 6 is the lest significant digit

- Each digit in a number has an associated weight.
- The LSD has a weight of 1, actually $10^0$
- The 0 is because we say the LSD is at position 0
- The 10 is because we are in base 10.
- (Remember $a^0 = 1, a \neq 0$)

- For numbers to the left of the LSB, the position increases by one each time we move to the left.
- In the number 7356
- 6 is at position 0 and has weight $10^0 = 1$
- 5 is at position 1 and has weight $10^1 = 10$
- 3 is at position 2 and has weight $10^2 = 100$
- 7 is at position 3 and has weight $10^3 = 1000$

- So the number 7356 is really $7\times10^3 + 3\times10^2 + 5\times10^1 + 6\times10^0$
- What are the digits past the third position?

- This works for numbers to the right of the decimal point as well.
- Consider 6.742
- 6 is at position 0 with weight $10^0$
- 7 is at position -1 with weight $10^{-1}$
- 4 is at position -2 with weight $10^{-2}$
- 2 is at position -3 with weight $10^{-3}$
- And the number is $6\times10^0 + 7\times10^{-1} + 4\times10^{-2} + 2\times10^{-3}$

- Why do you think we use base 10?
- Exercise: For a given number
- Give the MSD and LSD
- provide the polynomial in terms of base and powers that the number represents.