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Decimals

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 Sub Topics The numbers we use in arithmetic are all Decimal numbers. Different number systems are defined using the remainders that numbers yield on division by base. A decimal is a number that is written using base 10 system. The 10 digits we use to represent the numbers in decimal system, that are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 form a set of possible remainders, we get dividing any number by 10. The position or place values in decimal number system are all powers of 10.

What are Decimals?

A decimal is a number written using base 10 place value system. Each place value is a power of 10, the place values increase in the order of right to left and 10 times the place value to its right.
A decimal number can be expressed in verbal, decimal and expanded form.

Example:

 Verbal Form Forty three and fifteen hundredths Decimal Form 43.15 Expanded Form 4 Tens + 3 Ones + 1 Tenths + 5 Hundredths   40.0   +     3.0   +     0.1     +      0.05

Decimal Notation

A dot '.' is used as decimal notation and it is also called decimal point. A decimal number is equivalent to a mixed fraction consisting of integer and fractional part. The decimal point in a decimal number separates the integer part from the fractional (decimal) part.

For example, the number 72.86 is read shortly as 'Seventy two point eight six'. The digits to the left of decimal point 72 form the integer part and the numbers to the left of it 86 (86 hundredths) form the decimal part. The digit 0 is written in front of the decimal point for decimal numbers less than 1 as in 0.572.

Decimal Place Value

The place value chart used for integers is extended to include decimal place values. Let us place the decimal number 85.743 in Decimal place value chart to write the different forms of the number.

 Word Form Eight five and seven hundred forty three thousandths Decimal Form 85.743 Expanded Form 8 Tens + 5 Ones + 7 Tenths + 4 Hundreds + 3 Thousands 80.0 + 5.0 + 0.7 + 0.04 + 0.003

Rounding Decimals

Rounding decimals is done in a way similar to rounding integer.
• If the immediate digit to the right is $\geq$ 5, one is added to the digit in the rounding place.
• If the immediate digit to the right is < 5, the digit in the rounding place is retained.

Solved Examples

Question 1: Round 125.7489 to the hundredth.
Solution:
The digit in the hundredth's place is 4 and the digit to its right is 8 (in thousandth's place) i.e. greater than 5.
Hence, the number is rounded to the hundredth as 125. 75

Question 2: Round 89.24 to the tenth.
Solution:
The digit in hundredth's place is 4 which is less than 5.
Hence, the number is rounded to the tenth as 89.2

For adding Decimals, the numbers are lined up aligning the places as done for adding the integers. Zeros are appended at decimal place if needed and the sum is found.

Let us add 25.8, 17.423 and 8.76
The numbers are lined up as follows
25. 800   (Zeros appended in Hundredths and Thousandths places)
17. 423
8. 760   (Zero appended in Thousandths place)
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51. 983    (The three numbers added to get the sum)
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In case of signed numbers, rules applicable to integers are applied to decimals as well.
• If the two numbers added are of the same sign, find the sum and take the common sign.
• If the two numbers added are of the opposite sign, find the difference and take the sign of the number whose absolute value is larger.
• While adding more than two numbers, group the numbers by sign, find the sum in each group and add these two numbers to get the sum of given numbers.

Examples:
4.5 + 7.82 = 12.32
-10.84 + (- 4.62) = -15.46
84.6 + (-12.78) = 71.82
6.02 - 2.48 + 3.25 - 0.25 - 1.46 = (6.02 + 3.25) + (-2.48 - 0.25 - 1.46) = 9.27 + (-4.19) = 5.08

Subtracting Decimals

Subtraction is done with decimals as done with integers. The rules applied for subtracting signed decimals are also same as those used for signed integers.

Examples:
1. 7.05 - (-4.85) = 7.05 + 4.85 = 11.9
2. - 13.72 - (+ 7.63) = - 13.72 - 7.63 = - 21.35
3. - 24.84 - (10.62) = -24.84 + 10.62 = -14.22

Multiplying Decimals

In order to multiply two decimal numbers, they have to be first multiplied by ignoring the decimal point. The decimal point is then placed, counting the total number of decimal places from the right.

Let us multiply 46.75 and 8.2

46.75  (The number has two decimal places)
x  8.2  (The number has one decimal place)
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9350
374000
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383.350  <-- The decimal is placed counting three places from the right end.
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Ignoring the trailing zero, we can write 46.75 x 8.2 = 383.35

The sign rules are applied as applied to multiplying integers.
1. 4.25 x 0.72 = 3.06              + times + = +
2. -2.43 x - 0.02 = 0.0486       - times - = +
3. 72.5 x -3  = -217.5              + times - = -