1.3 Decimals

Decimals locate values between integers on a number line. Between $2.2$ and $2.3$, ten equal intervals represent hundredths, so $2.28$ is eight hundredths after $2.2$.

Trailing zeros do not change value: $5.2=5.20$. A zero between non-zero digits can hold a place, so $5.02$ is not equal to $5.2$.

Expanded form makes the values visible: $4.7634=4+0.7+0.06+0.003+0.0004$.

Zero holds an empty place. In $0.407$, the zero in the hundredths place keeps the $7$ in the thousandths place.

Comparing thousandths columns after alignment gives $0.078<0.080<0.700<0.707<0.870$, so the original numbers are ordered $0.078,0.08,0.7,0.707,0.87$.

For negative decimals, values farther left on the number line are smaller: $-0.8<-0.35$. Compare their positive sizes, then reverse the order because both are negative.

For $0.017$, there are three decimal places, so $0.017=\frac{17}{1000}$. The numerator and denominator have no common factor greater than $1$.

A decimal below $1$ becomes a proper fraction. A decimal such as $0.6$ becomes $60\%$, which is also below $100\%$.

For example, $0.407=\frac4{10}+\frac0{100}+\frac7{1000}=\frac{407}{1000}$. This is an exact equality, not an approximation.

Simplifying changes the name of the fraction, not the value. Thus $0.65$, $\frac{65}{100}$ and $\frac{13}{20}$ are the same number.

If the repeating block begins immediately, use $x$ and a power-of-ten multiple. For $x=0.4545\ldots$, $100x-x=45$, so $99x=45$ and $x=\frac5{11}$.

The multiplier must shift by the length of the repeating block, while a second multiplier may be needed to pass any non-recurring digits. The two decimals subtracted must have identical infinite tails.