1. This document discusses decimal numbers and operations involving decimals. It covers place value of decimals, ordering and comparing decimals, rounding decimals, and how to perform the basic math operations of addition, subtraction, multiplication, and division on decimals. It also discusses approximating square roots of decimals.

1. Unit 05 January
1. DECIMAL NUMBERS. PLACE VALUE.
1.1. PLACE VALUE.
Decimal Numbers are widely used in everyday life. We see them frequently in
money and in measurements of length, area, weight and so on. To express smaller
quantities than the unit, we use decimal numbers.
The value of a digit in a decimal depends on its position, or place, in the
decimal. Each place is 10 times the value of the next place to its right.
To read a decimal less than 1:
• Start at the decimal point.
• Read the number as a whole number. Then say the name of the place.
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2. Unit 05 January
To read a decimal greater than 1:
Do not get confused when naming numbers:
MATH VOCABULARY: Decimal Numbers, Tenths, Hundredths, Thousandths, Ten
Thousandths, Place Value, Decimal Point, Decimal Part, Whole Part, Millionths.
1.2. ORDERING DECIMAL NUMBERS.
Just like whole numbers, decimal numbers can be shown in a Number Line. To
do this we generally divide each segment of the number line in ten equal parts.
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3. Unit 05 January
Place the values 2.1, 2.45 and 2.8 on a number line.
Divide a number line from 2 to 3 into ten equal parts.
2.1 < 2.45 < 2.8
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4. Unit 05 January
We also have negative decimal numbers that can be represented in the
number line:
To compare decimal numbers without drawing the number line, we compare
the whole part, the number with the highest whole part will be the greatest one.
Compare −3.1; 2.45 𝑎𝑎𝑎𝑎𝑎𝑎 4.5
We compare the whole part of the numbers:
−3 < 2 < 4
Therefore:
−3.1 < 2.45 < 4.5
If the decimal numbers have the same whole part, we write them with the
same number of places after the decimal point, and then compare the decimal part,
the number with the highest decimal part will be the greatest one.
Compare 2.455; 2.45; 2.4
Firstly we write the numbers with the same number of places by writing zeros at the
end of decimal numbers without changing the place value of the other digits.
2.455; 2.450; 2.400
We compare the decimal part of the numbers:
400 < 450 < 455
Therefore:
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5. Unit 05 January
2.4 < 2.45 < 2.455
We have an important property of the decimal numbers; between two decimal
numbers we can always find other decimal numbers.
1.3. DECIMAL APPROXIMATIONS.
We are often given measurements as decimal numbers. In such cases we
approximate the decimal by Rounding Off to the required accuracy. We have previously
seen how to round off whole numbers. We round off decimal numbers in the same way.
Round off this number: 0.2382
0.2382 ≈ 0.238 (to 3 decimal places)
0.2382 ≈ 0.24 (to 2 decimal places)
0.2382 ≈ 0.2 (to 1 decimal place)
The rules for rounding off decimal numbers:
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• If the digit after the one being rounded is less than 𝟓𝟓, then we Round Down.
• If the digit after the one being rounded is 𝟓𝟓 or more, then we Round Up.
In English there is another way of rounding off decimal numbers, called to
Round to Significant Figures. That happens when we have some zeros before the first
non-zero figures.
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MATH VOCABULARY: To Round Off, Accuracy, To Round Down, To Round Up,
Significant Figures.
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2. ADDING AND SUBTRACTING DECIMALS.
When Adding or Subtracting decimal numbers, we write the numbers under
one another so the decimal points are directly underneath each other. When this is
done, the digits in each place value will also lie under one another. We then add or
subtract as for whole numbers.
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9. Unit 05 January
3. MULTIPLYING DECIMAL NUMBERS.
To Multiply decimals:
• Multiply as you would multiply whole numbers.
• Write the product.
• Then count the number of decimal places in both factors.
• Mark off that number of decimal places in the product.
MATH VOCABULARY: To Count, To Line Up.
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10. Unit 05 January
4. DIVIDING DECIMAL NUMBERS.
4.1. WHOLE NUMBERS WITH DECIMAL QUOTIENT
When we divide two whole numbers, many times we have non-exact divisions.
But we can continue dividing using decimal numbers. We just add the decimal point
once we reach the natural remainder.
4.2. DIVIDING A DECIMAL BY A WHOLE NUMBER.
To Divide a decimal by a whole number:
• Divide as you would with whole numbers.
• Place the decimal point in the quotient in the same place as the dividend.
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11. Unit 05 January
4.3. DIVIDING BY A DECIMAL.
To Divide by a decimal:
• Change the divisor to a whole number by moving the decimal point the
necessary number of places to the right.
• Move the decimal point in the dividend the same number of places to the right.
(Remember: Multiplying or dividing both numbers in a division by the same
factor does not change the result).
• Divide as you would with a decimal by a whole number.
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5. SQUARE ROOT AND DECIMAL NUMBERS.
The Square Root is also applying in decimal numbers:
But usually this numbers don´t have exact root, for this reason we use to work
with approximations:
√6.4 = �2 → 22
= 4 < 6.4
3 → 32
= 9 > 6.4
Usually we use the calculator and then we round off to the decimal place we
need. But we can also use the root algorithm that we learned on Unit 02.
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