什么是小数？

(Video) Fractions and Decimals

by Math Antics

Here's a video by mathantics that explains the basics of decimals.

$\large{\text{Summary}}$

A decimal is used to represent numbers between two integers. The following are examples of decimals. $$0.5,\quad\quad4.245,\quad\quad11.0,\quad\quad-5.001$$

With whole number place values, we multiply by $10$ to get to the next place value. Ones multiplied by $10$ gets us to tens. Multiply that by $10$ and we get hundreds. With decimals, we go the opposite direction and divide by $10$. Ones divided by $10$ gives us tenths, or $\frac1{10}$. Divide this by $10$ again and we get hundredths, or $\frac{1}{100}$.

The building blocks for a number, now that we include decimals, are: $$\cdots\quad1{,}000\quad100\quad10\quad1\quad\frac1{10}\quad\frac1{100}\quad\frac1{1{,}000}\quad\cdots$$

The decimal point is used to indicate we have a decimal number. It will always be between the ones place and the tenths place. Everything to the left of the decimal point is called the integer part of the number. Everything to the right of the decimal point is called the decimal part of the number.

How Do We Define "Decimal"?

Decimal actually has more than one meaning:

• A number that contains a so-called decimal point;
• A number written in base-$10$ system.

Here, we use decimal in the first sense: a number that contains a decimal point. Decimal point is the little point in $0.618$ that separates $0$ and $618$. When you write it down, the decimal point looks just like a period.

More Examples of Decimals

$-72, 1, 153, 496, 1729, 4294967297$ are not decimals.

However, $6.67408, 22.4, -0. \overline{142857}, 2.71828...$ are all decimals.

Notation

• the over-line denotes a repeating decimal here, meaning that the number $-0. \overline{142857}$ would actually look like $-0.142857142857142857142857142857142857142857142857...$ and it keeps repeating without stop.
• "..." means that the digits that follows are omitted.

What Do The Digits Mean?

In base-10 integers, each digit in the integer represent a power of $10$. The digits before the decimal point in a decimal, for example the "$3$" in "$3.14159$", represent the same number as the integer $3$.

In a base-10 integer, each digit carries $10$ times as much value as the digit after it, while the last digit represents how many "$1$"s there are. In digits after the decimal point, the first one represents the number of $\frac{1}{10}$s, the next one represents the number of $\frac{1}{100}$s, and each successive digit represents a smaller value.

For example, say we have the decimal: $$13.520229$$ Each digit has the following place value. $$\begin{array}{c|l} 1&\text{tens}\\ 3&\text{ones}\\ .&\text{decimal}\\ 5&\text{tenths, }\frac1{10}\\ 2&\text{hundredths, }\frac1{100}\\ 0&\text{thousandths, }\frac1{1{,}000}\\ 2&\text{ten thousandths, }\frac1{10{,}000}\\ 2&\text{hundred thousandths, }\frac1{100{,}000}\\ 9&\text{millionths, }\frac1{1{,}000{,}000} \end{array}$$

What is a Decimal?

A decimal is a number that contains a decimal point.

The number will be separated into two parts: the whole number part and the decimal part. These two parts are separated by the decimal point. Just like regular whole numbers, we can add, subtract, multiply, and divide decimals. They're numbers all the same.

Examples of Decimals: $$-1.2, \quad2.095,\quad 3.00$$

Decimals can represent both integers (like $3.00$) and non-integers (like $-1.2$).

Decimals are based on a powers of $10$ system and appear everywhere in scientific notation. As we move from left to right, the value of the digits gets divided by $10$.

• As we move to the left, each digit is $10$ times larger.
• As we move to the right, each digit is $10$ times smaller.

Any whole number can be written as a decimal (Example: $7.00, 19.00$).

A decimal can have an infinite amount of digits to the right of the decimal point. There are also an infinite number of decimals between any adjacent whole numbers on the number line.

We are now going to take a look at some of the types of problems you might encounter about the decimals.

Decimals are numbers that are separated into two parts (integer and decimal/fractional part) by a decimal point. Therefore, $1.57, 3.00, 0.12$ are examples of decimals.

Looking at the integer portion of the decimal, $25.67$ and $25.68$ are the largest.

Now looking at the decimal part of the number, $25.68$ is larger, so $25.68$ is the largest.