Category: 小数
What is Decimal
So, what is decimal?
Decimal is just another way to write fraction (or mixed number).
Decimal has two parts: integer part and fractional part (also called decimal part). Here you can see strong analogy with mixed number.
Decimals Place Value
So, what is a place value of a decimal?
Just like in a whole number value of a digit in a decimal depends on its place (position) in the number (before or after the decimal point).
The rightmost digit to the left of a decimal point is ones, next to the left tens, next hundreds, then thousands, ten thousands, hundred thousands, millions, ten millions; etc.
Rounding Decimals
Sometimes we need to round decimals.
Rounding decimals is very similar (almost identical) to rounding whole numbers.
Also, we need to know place value of decimals.
Example 1. Round 236.45 to the nearest ten.
Decimal Number Line
So, where are decimals located on a number line?
For example, where is 2.57 located? We see that this number has integer part 2 and decimal part 0.57.
This means that it is greater than 2 (because there is 2 and something else), but it is less than 2+1=3 because decimal part is always less than 1. Again, we see strong analogy with mixed numbers.
Comparing Decimals
Decimals are compared in nearly same way as integers.
Steps for comparing decimals:

Find number of digits in integer part of each number. If number of digits is not equal, add required number of LEADING zeros (just like in case of integers).
Powers of 10
When we talked about exponents, we said that raising $$$a$$$ to $$$b$$$th power is $$$a^b=\underbrace{a\cdot a\cdot a\cdot a\cdot...\cdot a}_{b}$$$.
$$$a$$$ is called base, $$$b$$$ is exponent (power).
Scientific Notation
In general, we need scientific notation, if we want to write very big or very small number more compactly.
For example, it is known that mass of the Earth is 5973600000000000000000000 kg. Look how long it is!
Decimal Fractions
Decimal fraction is a fraction, where denominator is a power of 10.
For example, $$$\frac{{1}}{{10}}$$$, $$$\frac{{17}}{{100}}$$$, $$$\frac{{3}}{{1000}}$$$ are all decimal fractions.
Nice thing about decimal fractions is that they can be easily converted into decimal.
Converting Decimals To Fractions
So, how to convert decimals to fractions?
It appears, that we have enough knowledge to do that.
Steps for converting decimals to fractions.
 It is known, that we can represent any number as number divided by 1: $$$a=\frac{a}{1}$$$. Do this.
 Simultaneously multiply numerator and denominator by 10, until you get a whole number in the numerator.
 Reduce the fraction.
Example 1. Convert 0.8 into fraction.
Converting Fractions to Decimals
Steps for converting fraction into a decimal:
 Find such integer number, that when multiplied by the denominator will give a power of 10 (10 or 100 or 1000 etc.)
 Multiply both numerator and denominator by that number (this can be done because of equivalence of fractions)
 As a result, we obtain a decimal fraction, that can be easily converted into a decimal.
As can be seen we use ease of converting decimal fraction and equivalence of fractions to convert arbitrary fraction.
Adding Decimals
Adding decimals is very similar to adding integers (more detailed examples in adding whole numbers note) except for only one key difference: you need to line up decimals (write dots one under another) before adding.
Subtracting Decimals
Subtracting decimals is very similar to subtracting integers (more detailed examples in subtracting whole numbers note) except for only one key difference: you need to line up decimals (write dots one under another) before subtracting.
Multiplying Decimals
Multiplying decimals is very similar to multiplying integers (more detailed examples in multiplying whole numbers note) except for only small difference: find number of digits to the right of the first number (number of decimal places), do the same for the second number. Multiply numbers, ignoring decimal points. Then put the decimal point in the answer; it should have as many decimal points as the first two found numbers combined.
Dividing Decimals by Whole Numbers
Dividing decimal by a whole number is very similar to dividing integers (more detailed examples in dividing whole numbers note). You just write decimal point and forget about it.
Example 1. Find $$${11.4}\div{2}$$$.
Dividing Whole Numbers by Decimals
Steps for dividing whole number by decimal:
 Simultaneously move decimal point to the right in both numbers, until you get whole number, instead of a decimal.
 Now, you have two whole numbers. Divide them, using long division, except that you need to continue division, until you get zero remainder.
Why this works?
Dividing Decimals
It is pretty easy to divide decimals after learning dividing decimals by whole numbers and dividing whole numbers by decimals.
To use techniques from these notes, simultaneously move decimal points to the right, until you get at least one whole number.
Repeating (Recurring) Decimals
Periodic (recurring) decimal is a decimal that has infinite number of digits, i.e. its digits repeat forever.
Until now, we converted fractions into decimals and divided decimals without a problem.
But there are situations, when we can't finish division.
Absolute Value
Absolute value makes any number positive.
On a number line we saw positive (to the right of zero) and negative (to the left of zero) numbers. There direction matters.
Absolute value doesn't care about direction. It only shows how far from zero number is.