CBSE Class 11 Applied Mathematics Number Theory Notes Set 01

Prime Numbers

Prime Number

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself.

Ramanujan Primes

The n^th Ramanujan prime \( R_n \) is the smallest integer for which there are at least n primes between \( \frac{x}{2} \) and x for all x ≥ \( R_n \). The first Ramanujan primes are 2, 11, 17, 29 and 41.
Note that the integer \( R_n \) is necessarily a prime number, because the number of primes between \( \frac{x}{2} \) and x can increase only by obtaining another prime at x = \( R_n \).

Ratio

Ratio

The value obtained when two similar quantities are compared by dividing one quantity by the other is called a ratio. The ratio of two quantities is that value which tells us how many times one quantity is of the other. Only quantities of the same kind, i.e., quantities with the same units, can be compared.

Notes on Ratio

1. The ratio of a and b is written as a : b and is measured by the fraction \( \frac{a}{b} \).
2. If two quantities are in the ratio a : b, then the first and the second quantities will be \( \frac{a}{a + b} \) times and \( \frac{b}{a + b} \) times the sum of the two quantities respectively.
Ex: If x : y = 3 : 5, then \( x = \frac{3}{8}(x + y) \) and \( y = \frac{5}{8}(x + y) \).
3. The ratio of two quantities can be found only when both the quantities are of the same kind.
Ex: The ratio between 1 metre and 5 seconds cannot be found, as the quantities given are not of the same kind.
4. A ratio is an abstract quantity and a ratio does not have any units.
Ex: The ratio of 30 seconds and one minute is 30 seconds : 60 seconds or 1 : 2.

Terms of a Ratio

For a given ratio a : b, we say that a is the first term or antecedent and b is the second term or consequent.
In the ratio 3 : 4, 3 is the antecedent while 4 is the consequent.

Properties of a Ratio

The value of a ratio remains the same if both the terms of the ratio are multiplied or divided by the same non-zero quantity. If a, b and m are non-zero real numbers, then
1. \( \frac{a}{b} = \frac{ma}{mb} \Rightarrow a : b = am : bm \)
2. \( \frac{a}{b} = \frac{a/m}{b/m} \Rightarrow a : b = \frac{a}{m} : \frac{b}{m} \)

Simplest Form of a Ratio

The ratio of two or more quantities is said to be in the simplest form if the highest common factor (HCF) of the quantities is 1. If the HCF of the quantities is not 1, then each quantity of the ratio is divided by the HCF to convert the ratio into its simplest form.

Example 1. Express 81 : 93 in its simplest form.
Answer:
The HCF of 81 and 93 is 3. We divide each term by 3.
Thus \( 81 : 93 = \frac{81}{3} : \frac{93}{3} = 27 : 31 \)
∴ The ratio 81 : 93 in its simplest form is 27 : 31.

Comparison of Ratios

Two ratios a : b and c : d can be compared in the following way: if \( \frac{a}{b} \) > \( \frac{c}{d} \), then a : b > c : d.

Example 2. Compare the ratios 4 : 5 and 18 : 25.
Answer:
Ratios can also be compared by reducing them to equivalent fractions with a common denominator.
\( 4 : 5 = \frac{4}{5} \) and \( 18 : 25 = \frac{18}{25} \)
\( \frac{4}{5} = \frac{4 \times 5}{5 \times 5} = \frac{20}{25} \)
As \( \frac{20}{25} > \frac{18}{25} \), i.e. \( \frac{4}{5} > \frac{18}{25} \), ∴ 4 : 5 > 18 : 25.

Types of Ratios

1. A ratio a : b, where a > b: if a positive quantity x is added to both terms of the ratio a : b, then (a + x) : (b + x) < a : b.
2. A ratio a : b, where a < b: if a positive quantity x is added to both terms of the ratio a : b, then (a + x) : (b + x) > a : b.
3. A ratio a : b, where a = b: if a positive quantity x is added to both terms of the ratio a : b, then (a + x) : (b + x) = a : b.
4. The compound ratio of two ratios a : b and c : d is ac : bd.

Example 3. Divide Rs. 780 among three friends P, Q and R in the ratio \( \frac{1}{4} : \frac{1}{3} : \frac{1}{2} \).
Answer:
Given ratio is \( \frac{1}{4} : \frac{1}{3} : \frac{1}{2} \). The LCM of the denominators is 12.
\( \Rightarrow \frac{1}{4} : \frac{1}{3} : \frac{1}{2} = \frac{1}{4} \times 12 : \frac{1}{3} \times 12 : \frac{1}{2} \times 12 = 3 : 4 : 6 \)
The sum of the terms of the ratio = 3 + 4 + 6 = 13
P's share = Rs. \( \left(780 \times \frac{3}{13}\right) \) = Rs. 180
Q's share = Rs. \( \left(780 \times \frac{4}{13}\right) \) = Rs. 240
And R's share = Rs. \( \left(780 \times \frac{6}{13}\right) \) = Rs. 360

Example 4. If x : y = 3 : 4 and y : z = 6 : 11, then find x : y : z.
Answer:
In both the ratios y is common. Try to make the value of y equal in both the ratios.
Multiply each term of the ratio 6 : 11 by \( \frac{2}{3} \) to make its first term (y) equal to 4.
\( \Rightarrow y : z = 6 \times \frac{2}{3} : 11 \times \frac{2}{3} = 4 : \frac{22}{3} \)
\( \therefore x : y : z = 3 : 4 : \frac{22}{3} = 9 : 12 : 22 \)

Example 5. In a bag there are coins in the denominations Rs. 1, Rs. 2 and Rs. 5 in the ratio 3 : 5 : 7 respectively. If the total value of the coins in the bag is Rs. 144, find the number of coins in each denomination and also the total value of Rs. 2 coins.
Answer:
Let the number of coins of denominations Rs. 1, Rs. 2 and Rs. 5 be 3x, 5x and 7x respectively. The total value of the coins in the bag
= Rs. [3x + 5x(2) + 7x(5)] = Rs. (3x + 10x + 35x) = Rs. 48x
Given, the total amount in the bag = Rs. 144
⇒ 48x = 144 ⇒ x = 3
The number of Rs. 1 coins in the bag = 3x = 9
The number of Rs. 2 coins in the bag = 5x = 5(3) = 15
The number of Rs. 5 coins in the bag = 7x = 21
∴ The total value of Rs. 2 coins in the bag = Rs. 15 × 2 = Rs. 30

Proportion

Proportion

The equality of two ratios is called a proportion.
Note: If a : b = c : d, then a, b, c and d are said to be in proportion and the same can be represented as a : b :: c : d.

Properties of Proportion

1. a, b, c and d are respectively known as the first, second, third and fourth proportionals.
2. The first and fourth terms are called extremes while the second and the third terms are called means.
3. Product of extremes = Product of means.
4. The fourth term can also be referred to as the fourth proportional of a, b and c.
5. If a : b = c : d, i.e., \( \frac{a}{b} = \frac{c}{d} \), then by taking the reciprocals of terms on both sides, b : a = d : c, i.e. \( \frac{b}{a} = \frac{d}{c} \). This relationship is known as ‘invertendo’.
6. If a : b :: c : d, then multiplying both sides of the proportion by \( \frac{b}{c} \), we get a : c = b : d. This relationship is known as ‘alternendo’.
7. Adding 1 to both sides of the proportion a : b :: c : d, we get \( \frac{a}{b} + 1 = \frac{c}{d} + 1 \Rightarrow \frac{a + b}{b} = \frac{c + d}{d} \) …(i)
That is, (a + b) : b = (c + d) : d. This relationship is known as ‘componendo’.
8. Subtracting 1 from both sides of the proportion a : b :: c : d, we get \( \frac{a}{b} - 1 = \frac{c}{d} - 1 \Rightarrow \frac{a - b}{b} = \frac{c - d}{d} \) …(ii)
That is, (a − b) : b = (c − d) : d. This relationship is known as ‘dividendo’.
9. Dividing equation (i) by equation (ii), we get \( \frac{a + b}{a - b} = \frac{c + d}{c - d} \), i.e. (a + b) : (a − b) = (c + d) : (c − d). This relationship is known as ‘componendo and dividendo’.
10. If \( \frac{a}{b} = \frac{c}{d} = \frac{e}{f} \) and l, m and n are any three non-zero numbers, then \( \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{la + mc + ne}{lb + md + nf} \).

Continued Proportion

Three quantities a, b and c are said to be in continued proportion if a : b :: b : c.
If a : b :: b : c, then c is called the third proportional of a and b.

Mean Proportional

If a : b :: b : c, then b is called the mean proportional of a and c.
We have already learnt that product of means = product of extremes.
⇒ b × b = a × c ⇒ \( b^2 = ac \) ⇒ \( b = \pm\sqrt{ac} \)
∴ The mean proportional of a and c is \( \sqrt{ac} \).

Example 6. If 4 taps can fill a tank in 10 hours, then in how many hours can 6 taps fill the same tank?
Answer:
Assume that 6 taps can do the same work in x hours. As more taps require fewer hours to do the same work, the number of taps varies inversely as the number of hours.
∴ (Inverse ratio of number of taps) :: (Ratio of number of hours)
⇒ 6 : 4 :: 10 : x
\( \Rightarrow \frac{6}{4} = \frac{10}{x} \Rightarrow x = \frac{10 \times 4}{6} = 6\frac{2}{3} \) hours
∴ Six taps can fill the tank in \( 6\frac{2}{3} \) hours.

Example 7. In a family, the consumption of power is 120 units for 18 days. Find how many units of power are consumed in 30 days.
Answer:
Let the consumption of power in 30 days be x units.
The more the period of time, the more is the consumption of power. Hence the consumption of power varies directly as the number of days.
∴ (Ratio of number of days) :: (Ratio of consumption of power)
⇒ 18 : 30 :: 120 : x
Product of means = Product of extremes ⇒ 30 × 120 = 18 × x
\( \Rightarrow x = \frac{30 \times 120}{18} = 200 \) units
∴ The consumption of power for 30 days is 200 units.

Example 8. 4 men, each working 6 hours per day, can build a wall in 9 days. How long will 6 men, each working 3 hours per day, take to finish the same work?
Answer:
Clearly, \( M \propto \frac{1}{D} \) and \( M \propto \frac{1}{H} \), where M = number of men, D = number of days and H = hours per day.
⇒ MDH = constant
\( \Rightarrow M_1D_1H_1 = M_2D_2H_2 \)
\( \Rightarrow 4 \times 9 \times 6 = 6 \times D_2 \times 3 \)
\( \Rightarrow D_2 = 12 \)
∴ Six men, each working 3 hours per day, can finish the job in 12 days.

Logarithms

Logarithm

Logarithms are useful in long calculations involving multiplication and division.

Definition

The logarithm of any positive number to a given base (a positive number not equal to 1) is the index of the power of the base which is equal to that number. If N and a (≠ 1) are any two positive real numbers and for some real number x, \( a^x = N \), then x is said to be the logarithm of N to the base a. It is written as \( \log_a N = x \), i.e. if \( a^x = N \), then \( x = \log_a N \).
Ex: \( 2^3 = 8 \Rightarrow 3 = \log_2 8 \).
From the definition of logarithms, we get the following results when a > 0, a ≠ 1 and b > 0:
1. \( \log_a a^n = n \)
2. \( a^{\log_a b} = b \)

Systems of Logarithms

There are two systems of logarithms, natural logarithms and common logarithms, which are used most often.
1. Natural logarithms: These were discovered by Napier. They are calculated to the base e, which is approximately equal to 2.718.
2. Common logarithms: Logarithms to the base 10 are known as common logarithms.

Properties of Logarithms

1. Logarithms are defined only for positive real numbers.
2. Logarithms are defined only for positive bases different from 1.
3. In \( \log_b a \), neither a nor b is negative, but the value of \( \log_b a \) can be negative. Ex: As \( 10^{-2} = 0.01 \), \( \log_{10} 0.01 = -2 \).
4. Logarithms of different numbers to the same base are different, i.e. if a ≠ b, then \( \log_m a \neq \log_m b \). In other words, if \( \log_m a = \log_m b \), then a = b.
5. Logarithms of the same number to different bases have different values, i.e. if m ≠ n, then \( \log_m a \neq \log_n a \) (a ≠ 1). In other words, if \( \log_m a = \log_n a \), then m = n.
6. Logarithm of 1 to any base is 0.
7. Logarithm of 0 is not defined.

Laws of Logarithms

1. \( \log_m (ab) = \log_m a + \log_m b \)
2. \( \log_m \left(\frac{a}{b}\right) = \log_m a - \log_m b \)
3. \( \log_m a^n = n \log_m a \)
4. \( \log_b a \cdot \log_c b = \log_c a \)
5. \( \log_b a = \frac{\log_c a}{\log_c b} \) (change of base). In this relation, if we take c = a, we get \( \log_b a = \frac{1}{\log_a b} \).

Logarithm of a Number to Base 10

Consider the numbers 2, 20, 200, 0.2 and 0.02. We see that 20 = 10 × 2 and 200 = 100 × 2.
∴ log 20 = 1 + log 2 and log 200 = 2 + log 2
Similarly, log 0.2 = −1 + log 2 and log 0.02 = −2 + log 2. From the table, we see that log 2 = 0.3010.
∴ log 20 = 1.3010, log 200 = 2.3010, log 0.2 = −1 + 0.3010 and log 0.02 = −2 + 0.3010.
We note two points:
1. Multiplying or dividing by a power of 10 changes only the integral part of the logarithm, not the fractional part.
2. For a number less than 1 (for example 0.2), it is more convenient to write the log value as −1 + 0.3010 instead of changing it to −0.6990.

Characteristic and Mantissa

For numbers less than 1, as the negative sign refers only to the integral part, it is written above the integral part rather than in front, i.e. \( \log 0.2 = \bar{1}.3010 \) and not −1.3010.
The integral part is called the characteristic and the fractional part (which is always positive) is called the mantissa.

Example 9. Express −0.5229 in the standard form.
Answer:
\( -0.5229 = -1 + 1 - 0.5229 = -1 + 0.4771 = \bar{1}.4771 \)

Rule to Obtain the Characteristic of log x

1. If x > 1 and there are n digits in the integral part of x, the characteristic is n − 1.
2. If x < 1 and there are m zeroes between the decimal point and the first non-zero digit of x, the characteristic is −(m + 1), more commonly written as \( \overline{m + 1} \). For example, the characteristic of log 0.2 is \( \bar{1} \) and that of log 0.025 is \( \bar{2} \).
Note: \( -4 = \bar{4} \), but \( -4.01 \neq \bar{4}.01 \).

Finding the Logarithm of a Number from the Log Tables

Example 10. Find the value of log 25 and log 0.025.
Answer:
In the log table, we find the number 25 in the left hand column. In this row, in the next column (under 0), we find 0.3979.
For the logarithm of all numbers whose significant digits are 25, this number 0.3979 is the mantissa. Prefixing the characteristic, we have
log 25 = 1.3979
log 250 = 2.3979
\( \log 0.025 = \bar{2}.3979 \)

Example 11. Find the value of log 2.546 and log 25460.
Answer:
In the log table we locate 25 in the first column. In this row, in the column under 4, we find 0.4048.
Since there are four significant digits in 2546, in the same row, under column 6 of the mean difference columns, we find the number 10. So the mantissa of the logarithm of 2546 will be 4048 + 10 = 4058.
Thus log 2.546 = 0.4058
\( \log 0.2546 = \bar{1}.4058 \)
log 25460 = 4.4058

Antilogarithm

As \( \log_2 8 = 3 \), 8 is the antilogarithm of 3 to the base 2. Antilog b to the base m is \( m^b \).
We saw that log 2.546 = 0.4058. Therefore antilog 0.4058 = 2.546.

Example 12. Find the antilog of 2.3246.
Answer:
We have to locate 0.32 in the left hand column of the antilog table, slide along the horizontal line and pick out the number in the vertical column headed by 4. We see that the number is 2109.
The mean difference for 6 in the same line is 3.
The significant digits in the required number are 2109 + 3 = 2112.
As the characteristic is 2, the integral part has 3 digits, so the required antilog is 211.2.

Example 13. If \( \log_{10} 3 = 0.4771 \) and \( \log_{10} 2 = 0.3010 \), find the value of \( \log_{10} 48 \).
Answer:
\( \log_{10} 48 = \log_{10} 16 + \log_{10} 3 \)
\( = \log_{10} 2^4 + \log_{10} 3 \)
\( = 4\log_{10} 2 + \log_{10} 3 \)
= 4(0.3010) + 0.4771
= 1.6811