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Naming decimal places

naming decimal places

Naming decimal places can be defined as an expression of place value in words.

Apart from integers whose value has to be greater than “negative infinity” (symbol: -\infty) and smaller than “positive infinity” or just “infinity” (symbol: +\infty), there are also decimal numbers that represent the number of equal portions between two adjacent integers (adjacent integers are numbers that have been placed before and after the representing number). A decimal number is made of an integer part, placed on the left side of a decimal point, and a fractional part, placed on the right side of a decimal point. As a matter a fact, decimals are numbers which tells us how many parts of a whole we have. We use them to mark measure units of things that are not completely whole.

decimal number

Decimals are numbers which tells us how many parts of a whole we have. We use them to mark measure units of things that are not completely whole.

unit, decimal point and mantissa of a decimal

Unit is a part of a decimal which tells us how many whole parts we have while mantissa tells us how many parts of a whole do we have left.

The decimal or a decimal number can be written as a fraction. As an example of a decimal number, let’s take the number 28.531. Now, number 28 (the left side of the decimal point) represents the integer part of the decimal number and .531 (the right side of the decimal point) represents the decimal part of the number. The .531 represents a value which is smaller than 1, but larger than 0 and can also be represented as a fraction \frac{531}{1000}. No matter how many digits there are after the decimal, their combined value is always less than 1 but more than 0. Thus, the value of the number \ 28.531 is greater than the value of the whole number 28, but lesser than the value of the whole number 29.

decimals as fractions

Naming decimal places plays an important role in the representation of the number as a whole. Since the decimal system we use is a positional numeric system, all of the digits in a decimal number is termed according to their position in respect to the decimal point and it is important to name the decimal places properly. The entire decimal system is completely based on number 10 and all of the digits, before and after the decimal point, are defined in terms of ten because of that.

The digit placed furthest to the right of the decimal point has the smallest value. Hence, in the number 28.531, the digit 5 is placed furthest from the decimal point and hence has the smallest value. The entire number can be defined as 2\cdot 10+8 \cdot 1+5 \cdot \frac{1}{10}+3 \cdot \frac{1}{100}+1 \cdot \frac{1}{1000}. The number after the decimal point can be collectively pronounced as 6 tenths, 4 hundredths and 5 thousandths or, more simply, as 645 thousandths.

place values

All the place values of the numbers depend on position on the left or right side of a decimal point. Look at the example with more digits.
Let’s take a look at, for example the number 1,987,654,321.123456,
The first digit before the decimal point represents the ones (number 1),
-the second stands for the tens (number 2), the third for the hundreds (number 3),
-the fourth for the thousands (number 4, after the comma),
-the fifth for the ten thousands (number 5),
-the sixth for the hundred thousands (number 6),
-the seventh for the millions (number 7, after the second comma),
-the eight for the ten millions (number 8),
-the ninth for the hundred millions (number 9) and
-the tenth for the billions (number 1, after the third comma).
All digits after the decimal point are called decimals.
-The first digit represents tenths (number 1),
-the second digit stands for the hundredths (number 2),
-third for the thousandths (number 3),
-fourth for the ten thousandths (number 4),
-fifth for the hundred thousandths (number 5),
-sixth for the millionths (number 6)

There are larger and smaller place values, but these ones are used the most. It doesn’t matter how large the number of digits is, they can be read and understood with ease. Test the knowledge with worksheets.

Addition and subtraction

Addition of decimal numbers is pretty much the same as the one with the whole numbers, only with a decimal point.

Example: Solve:

1.5555 + 1.7

Write one beneath the other, in a way that the decimal points align. This is the most important step; this is how you know which parts you’re adding.

As in any other addition you start from the left, and if you gain more than ten you simply transfer one to the other side. The same goes for the subtraction.

addition and subtraction decimal numbers

Multiplication

2.56 \cdot 1.5

First step in multiplication of two decimal numbers is to multiply each one of them with 10, 100, 1000 and so on to get whole numbers.

For our example that would be 256 \cdot 15
And this is something we know 2568 \cdot 150 = 3840
The last step is to put the decimal point in place. Where would that be? This depends on the number of elements in mantissas of numbers you’re multiplying; their sum will be the number of elements in mantissa in their product.

2.56 – number of elements in mantissa  is 2

1.5 – number of elements in mantissa is 1

2.56 \cdot 1.5 – numbers of elements in mantissa is 3.

This means that in the number 3870 we have to put the decimal point in the third place from the right.

This leads us to our final solution:

2.56 \cdot 1.5 = 3.84 (the last zero can be disregarded)
Second way to do the multiplication:

You multiply as you always multiplied, but just when you finish take down the decimal point:

multiplying decimal numbers

Division

Division of two decimal numbers is similar to division of two whole numbers; but there are few changes.

The quotient won’t change if you multiply each number with the same number. This means that you can transform your decimal numbers into whole numbers, and with them you already know how to calculate.

Example 1:
First you multiply it with 10, 100, 1000 to get both numbers whole. In this example we’ll multiply both numbers with number 1000:

2.514 : 1.257
2514 : 1257 = 2 and this is your solution.

Example 2:

2.5 : 1.25 = ?

2.5\cdot 100

1.25\cdot 100
250 : 125 = 2

Example 3:

How about when you have two whole numbers, but divisor is greater than dividend?

1 : 2 = 0

As you already know, one does not contain any two’s. this means that our decimal number will be something in a form of 0._.

This is the point where you put a decimal place behind that zero, and then you simply add zeros to the left side and continue your division. We can do that because we know that every whole number can be written as a unit and infinitely many zeros behind the decimal point, which means that our one becomes 1.0000 as many zeros we need.

1 : 2 = 0.5 (now one zero comes down)

dividing decimal numbers

Example 4:

3 : 4 = ?

dividing decimals part 2

Let’s explain decimals in an example.


You’re eating in a restaurant and order a pie. Now you have one whole pie.

whole pie

 

How many pies do you have if you eat one half?

half of a pie

 

As you know you have one half, or \frac{1}{2} pie, now you have to transform it into a decimal. Since we learned that this should be easy.

You have 0.5 pie.

And if you buy another pie, how many pies in decimals do you have?

one pie plus half pie

Now you have \ 1 + 0.5 = 1.5 pies.


Let’s remember how we could write any whole number using decomposition in thousands, hundreds, tens and ones.

For example number 2 554 can be written as:

2 554 = 2000 + 500 +50 + 4

Which means that number 2 554 contains two thousands, 5 hundreds, 5 tens and 4 ones.

What if we try to do that with decimals?

For example, number 3.14 can be represented as \ 3 \cdot 1 + \frac{1}{10} + \frac{4}{100}


That is the rule:

addition of decimals rule

Considering this, we can manipulate decimal numbers in any way we want.

Number 3.14 can also be written as:

  3\cdot 1 + \frac{14}{100},

\frac{314}{1000},

\frac{3140}{10000}


How do you do that?

Any decimal number can be written as a fraction whose numerator and denominator are whole numbers. The easiest way to remember this is: as many decimal places does your number have, that’s how many zeros in a denominator you’ll have:

3.14 = \frac{314}{100}

3.145 = \frac{3145}{1000}

31.5 = \frac{315}{10}

512.512 = \frac{512}{1000}


Comparing decimals

One decimal is greater than the other if one has greater value.

How would you know which one is greater?

You simply go by the decimals and the first different digit will tell you. If that digit is greater than the other one, than that whole number is greater.

Example: Compare two numbers.

A = 1.23457
B = 1.23456

You go digit one by one and see that first five digits are the same. But sixth digit in number A is greater than the sixth digit in number B. that means that

A > B.

Example 2:

C = 2.12345678954545
D = 1. 12345678954545

Here we have no problem, because the numbers differ in the first digit which means that C > D

Example 3:

E = 12.35478
F = 1.235478

On first look you might thing these two numbers are the same, but be careful about the position of the decimal point. E > F.

Decimals on the number line

Decimal numbers are, just like whole numbers, divided on the positive ones, and negative ones.

Positive decimal numbers are found on the right side of the point of origin, and negative ones on the left.

Between any two numbers on the number line lies infinitely many decimal numbers.

The safest way to be precise about placing a decimal number on the number line is to convert it into a fraction.

Example:

Place number 0.25 on the number line.

As we already learned we can transform this number into a fraction:

0.25 = \frac{25}{100}

And we can shorten this fraction into \frac{1}{4}.

This means that this point is \frac{1}{4} away from the point of origin to the right. First we’ll divide our segment from 0 to 1 into four parts and take the first dot.

Example 2:

Place number 1.2 on the number line.

We’ll again transform it into a fraction: 1.2 = \frac{12}{10} = \frac{6}{5} = \frac{11}{5}

This means that our number is located between 1 and 2, \frac{1}{5} away from 1.

Example 3:

Place number - 2.45 on a number line.

-2.45 = - \frac{245}{100} = - \frac{49}{20} = - 1\frac{29}{20}

For numbers with many decimals or decimal number that are not obvious, like fractions \frac{1}{2} , \frac{1}{4} and so on, you can use approximated place. For example, this number is very close to -2,5 or a fraction -2 \frac{1}{2} so we’ll draw it close to it, but slightly to the right, because number -2.45 > -2.5.

Naming decimal places worksheets

Naming decimal places

  Ones - tens - hundreds - thousands (79.6 KiB, 1,008 hits)

  Thousands - ten thousands - hundred thousands - millions (78.7 KiB, 796 hits)

  Millions - ten millions - hundred millions - billions (83.2 KiB, 774 hits)

  Ones - tenths - hundredths - thousandths (76.7 KiB, 1,035 hits)

  Thousandths - ten thousandths - hundred thousandths - millionths (77.4 KiB, 725 hits)

  Millionths - billionths (151.3 KiB, 718 hits)

Addition

  Two positive decimals (178.5 KiB, 668 hits)

  Three positive decimals (207.1 KiB, 583 hits)

  Four positive decimals (244.3 KiB, 639 hits)

  Two decimals (240.5 KiB, 624 hits)

  Three decimals (296.2 KiB, 557 hits)

  Four decimals (352.7 KiB, 559 hits)

Subtractions

  Two positive decimals (84.6 KiB, 556 hits)

  Three positive decimals (97.7 KiB, 536 hits)

  Four positive decimals (105.8 KiB, 517 hits)

  Two decimals (109.5 KiB, 554 hits)

  Three decimals (130.8 KiB, 515 hits)

  Four decimals (152.6 KiB, 620 hits)

Multiplication

  Two positive decimals (59.0 KiB, 542 hits)

  Three positive decimals (64.1 KiB, 514 hits)

  Four positive decimals (68.1 KiB, 539 hits)

  Two decimals (88.9 KiB, 582 hits)

  Three decimals (104.6 KiB, 511 hits)

  Four decimals (121.0 KiB, 509 hits)

Division

  Two positive decimals (77.4 KiB, 635 hits)

  Two decimals (130.0 KiB, 599 hits)

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