Another Peek Beyond the Point Class 7 Notes

Another Peek Beyond the Point Class 7 Notes

Another Peek Beyond the Point Class 7 Notes

1. Decimals

Decimals express fractions in a base-10 system, allowing representation of non-whole numbers.

Example: 0.5 = [latex]\frac {1}{2}[/latex], 2.75 = [latex]\frac {11}{4}[/latex]

Place Value System

2. Decimal Multiplication Process:

1. Multiply as whole numbers ignoring decimal points.
2. Count total decimal places in factors.
3. Place the decimal point in the product.

The product of two decimals can vary in relation to the numbers multiplied.

Relationship between the product and the numbers multiplied :

3. Decimal Division

Expressing Fractions as Decimals :

Use long division or find equivalent fractions with denominators of (10), (100), etc.

Division Using Place Value :

Perform long division as with whole numbers, keeping track of the decimal point.

Converting Divisors :

When dividing by a decimal, multiply both the dividend and divisor by the same power of (10).

4. Repeating Decimals

Some divisions result in repeating decimals.

Example : (10 = 3 = 3.333...)

5. Cyclic Numbers Definition : A cyclic number is a number whose multiples are cyclic permutations of its digits.

Example :

(142857) :

(142857 × l = 142857)

(142857 × 2 = 285714)

(142857 × 3 = 428571)

CHAPTER AT A GLANCE

1. Decimal : A decimal is a way of writing numbers that are not whole. It uses a dot (called a decimal

point) to separate the whole number part from the fractional part. For example, in the number 27.53, the 27 is the whole number, and .53 represents the fraction. .

2. Place Value : This refers to the value of a digit based on its position in a number. In the number 27.53:

• The 2 is in the tens place (20).

• The 7 is in the ones place (7).

• The 5 is in the tenths place (0.5).

• The 3 is in the hundredths place (0.03).

3. Fraction : A fraction represents a part of a whole and is written as one number over another, like —. The top number is called the numerator, and the bottom number is the denominator.

4. Equivalent Fractions: These are different fractions that represent the same value. For example, [latex]\frac {1}{2}[/latex] is equivalent to [latex]\frac {2}{4}[/latex].

5. Multiplying Decimals : When you multiply decimals, you first multiply the numbers as if they were whole numbers. Then, you count the total number of decimal places in both numbers and place the decimal point in the product accordingly.

6. Dividing Decimals : To divide decimals, you can use long division. If the divisor (the number you are dividing by) is a decimal, you can move the decimal point to make it a whole number, and then move the decimal point in the dividend (the number being divided) the same number of places.

7. Rounding : This is the process of adjusting a number to make it simpler but keeping it close to the original value. For example, rounding 4.6 to the nearest whole number gives you 5.

8. Quotient: The result of dividing one number by another. For example, in the division 12 ÷ 3 = 4, the number 4 is the quotient.

9. Recurring Decimal : A decimal that has a digit or group of digits that repeat infinitely. For example, 0.333... (where 3 repeats forever) is a recurring decimal

A Quick Recap of Decimals

Decimals are a way of expressing fractions in a base-10 system. They are particularly useful for representing numbers that are not whole, allowing us to express values that fall between integers. For example,' the decimal 0.5 represents the fraction [latex]\frac {1}{2}[/latex], and 2.75represents the fraction [latex]\frac {11}{4}[/latex].

Place Value System

1. Division is essentially the process of deter-deter¬mining how many times one number (the divisor) fits into another number (the dividend).
2. The result of a division is called the quotient. For example, in the division 128 * 4 = 32, (128) is the dividend, (4) is the divisor, and (32) is the quotient.

Decimal Multiplication

When multiplying decimals, the process is similar to multiplying whole numbers. However, we must pay to the placement of the decimal point in the final product.

Example: To multiply 5.8 and 1.24:

1. Ignore the decimal points and multiply as if they were whole numbers : 58 × 124 = 7192

2. Count the total number of decimal places in the factors,:

1. 5.8 has 1 decimal place.
2. 1.24 has 2 decimal places.
3. Total 1 + 2= 3 decimal places.

3. Place the decimal point in the product: 5.8 × 1.24 = 7.192 .

Is the Product Always Greater than the Numbers Multiplied?

The product of two decimals can vary in relation to the numbers multiplied. Relationship between the product and the numbers multiplied:

Decimal Division

Dividing decimals can be straightforward if we understand how to manipulate them correctly. Expressing a Fraction as a Decimal It is easy to express a fraction as a decimal if the denominator is a power of 10 (1, 10, 100, 1000, etc.).

Example : To convert the fraction [latex]\frac {29}{10}[/latex] to a decimal, we can find an equivalent fraction with a denominator of 100. Since 4 × 25 = 100, we multiply both the numerator and denominator by 25: [latex]\frac{29 \times 25}{4 \times 25}[/latex] = [latex]\frac {725}{100}[/latex] = 7.25

To convert fractions to decimals, you can perform long division or find an equivalent fraction with a denominator of (10), (100), etc.

Division Using Place Value

Long division with decimals involves dividing as we would with whole numbers, but we must keep track of the decimal point.

When dividing by a decimal, you can convert the divisor to a whole number by multiplying both the dividend and the divisor by the same power of 10.

Example : To divide 126 by 2.5 :

Convert 2.5 to a fraction :

126 ÷ 2.5 = [latex]\frac {25}{10}[/latex] = 126 × [latex]\frac {10}{25}[/latex] = [latex]\frac {1260}{25}[/latex]

Now perform long division:

1. 25 goes into 1260 a total of 50 times with a remainder that leads to a decimal.
2. The final result is 50.4.

Division with a Decimal Quotient

When dividing by a decimal, we can convert the divisor to a whole number by multiplying both the dividend and divisor by the same power of 10.

Example : To find 4.68 ÷ 1.3:

Multiply both by 10 to eliminate the decimal:

46.8 ÷ 13 = [latex]\frac {46.8}{13}[/latex]

Division with a Decimal Dividend

When the dividend is a decimal, we can still perform the division as usual.

Example : To find 0.75 ÷ 0.25 :

Convert to fraction : [latex]\frac {0.75}{0.25}[/latex] = [latex]\frac{\frac{75}{100}}{\frac{25}{100}}[/latex] = [latex]\frac {75}{25}[/latex] = 3.

Division with a Decimal Divisor

When the divisor is a decimal, we can convert it to a whole number by multiplying both the dividend and divisor by the same power of 10.

Example : To find 4.68 ÷ 0.13 :

Multiply both by 100 : 4.68 ÷ 0.13 = [latex]\frac {468}{13}[/latex]. 

Does This Ever End?

Some decimal divisions result in repeating decimals. For instance, dividing 1 by 3 gives 0.333..., which continues indefinitely.

A Magic Number : 142857

The number 142857 is known as a cyclic number. When multiplied by 1, 2, 3, 4, 5, 6, the products are cyclic permutations of the digits of 142857. For example:

1. 142857 × 2 = 285714
2. 142857 × 3 = 428571

This property is fascinating and leads to deeper explorations in number theory;

Dividend, Divisor, and Quotient

In division, the relationship between these three components is crucial:

1. Dividend : The number being divided.
2. Divisor : The number by which we divide.
3. Quotient: The result of the 'division.

For example, in 128 ÷ 4 = 32 :

1. Dividend = 128
2. Divisor = 4
3. Quotient = 32