## The Decimal System

The decimal system is a way of representing a number using powers of ten. The first digit before the decimal is the *ones* place. Suppose we have a digit, $ d $, that is in the ones place. Since $ d $ is in the ones place the value of $ d = d \times 10^0. $ We use 10 since we are in the decimal system (base 10). We use 0 for the exponent because we are 1 digit to the left of the decimal. Why 0? In Computer Science and Mathematics, counting is often started at 0. Also $ 10^0 = 1$ which is the value that the digit in the ones place is multiplied by.

What is the value of 93? We find the value of 93 by adding the value of the digits in the ones and tens place together.

$$ 93 = 9 \cdot 10^1 + 3 \cdot 10^0 $$

The value of the *hundreds* place is found by multiplying by $10^2 = 100.$

What about the places to the right of the decimal? The first place to the right of the decimal is the *tenths* place. Whatever digit is in the tenths place we multiply by $\frac{1}{10}$ to get its value. So the value of $0.7 = 7 \times 10^{-1} = 7 \times \frac{1}{10}.$ Numbers with nonzero digits to the right of the decimal are fractional components.

The digit two places to the right of the decimal is the *hundredths* place. To get the value of that digit we multiply by $\frac{1}{100}.$ So for the number 0.07, the zeroes carry no value, but the 7 in the hundredths place means the number carries the value of $ \frac{7}{100} = 7 \times 10^{-2}$ . The digit three places to the right of the decimal is the *thousandths* place. The digit in the thousandths place is multiplied by $ \frac{1}{1000} = 0.001.$

Each digit to the right of the decimal carries a weight that is 10 times less than the digit before it, and each digit to the left of the decimal place carries a value ten times greater than the digit before it (assuming the digits are the same, of course.)