Introduction To Decimals: 동영상 및 연습문제

The place value system for decimals is based on powers of 10, similar to whole numbers. For whole numbers, digits to the left of the decimal point represent ones, tens, hundreds, and so on, increasing by powers of 10 as you move left. For decimals, digits to the right of the decimal point represent fractional parts such as tenths, hundredths, thousandths, decreasing by powers of 10 as you move right. For example, in the decimal number $986.071$, the digit 9 is in the hundreds place, 8 in the tens place, 6 in the ones place, 0 in the tenths place, 7 in the hundredths place, and 1 in the thousandths place. This system helps us understand the value of each digit depending on its position relative to the decimal point.

To read and write decimals in words, first separate the whole number part from the decimal part. The decimal point is read as "and." For example, the decimal $3.14$ is read as "three and fourteen hundredths." The name of the last digit's place value determines how the decimal part is named. In this case, the last digit 4 is in the hundredths place, so we say "fourteen hundredths." For a number like $12.032$, it is read as "twelve and thirty-two thousandths" because the last digit 2 is in the thousandths place. This method helps clearly express decimals in words by linking the digits to their place values.

Decimals can be converted into fractions by using the digits to the right of the decimal point as the numerator and a denominator based on the place value of the last digit. For example, the decimal $0.25$ has digits 25 after the decimal, and since the last digit 5 is in the hundredths place, the denominator is 100, so the fraction is $\frac{25}{100}$. This fraction can be simplified to $\frac{1}{4}$. If there is a whole number part, like in $8.4$, write it as a mixed number: the whole number 8 plus the fraction $\frac{4}{10}$, which simplifies to $\frac{2}{5}$. So, $8.4$ becomes $8\frac{2}{5}$.

Decimals and fractions both represent parts of a whole. A decimal is another way to express fractions, especially those with denominators that are powers of 10, such as 10, 100, or 1000. For example, the fraction $\frac{3}{10}$ is equivalent to the decimal $0.3$. Decimals use place values to show parts of a whole, with digits to the right of the decimal point representing tenths, hundredths, thousandths, etc. Understanding this relationship allows you to convert decimals to fractions by identifying the place value of the last digit and using it as the denominator, and vice versa.

To simplify fractions obtained from decimals, first write the decimal as a fraction with the numerator as the digits after the decimal point and the denominator as a power of 10 based on the place value of the last digit. Then, find the greatest common factor (GCF) of the numerator and denominator and divide both by this number. For example, for the decimal $0.25$, the fraction is $\frac{25}{100}$. The GCF of 25 and 100 is 25, so dividing numerator and denominator by 25 gives $\frac{1}{4}$, which is the simplified fraction. Simplifying fractions makes them easier to understand and work with in calculations.