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Demystified Foundations of University Mathematics_M.B. Abdullahi

1.0.Laws of Indices

These are laws or axioms governing indices, the plural of index. For example, 2 is the base and 3 is the index of . In , b and c are the indices of a. The indices of a in are and respectively. So indices may be negative, rational, real or complex. They may be any existing number.

Many preliminary topics depend on this topic. This is the reason why many texts begin with it. The following are the laws governing indices.

(a) Laws of Indices

i. ;

ii. ;

iii. ;

iv. ;

v. ;

vi. ; and

vii. .

Examples

(1) If

Solution:

by law i.

Thus, , whence.

by dividing both sides.

since 1 equals to 2 power 0 by law iii.

Hence, x = 0.

(2) Prove that

Proof: Consider LHS

Since this is , we have:

Also,

Conversely,

This completes the proof.

(3) Solve.

Solution: Since we have: .

By taking LCM, we get:

We are dealing with , hence, we have: finally.

(4) Solve

Solution:

Given

Using first law

Factoring out common terms

Dividing both sides

Finally

(5) Solve

Solution: Hence, we will get: .

Dividing both sides by we get:

by equating indices.

(6) Prove that and show that.

Proof: We prove this by contradiction. Squaring both sides, we have: . This means

That is,

This is plainly false. Hence, .

(7) Solve .

Solution: Law iv.

Hence, by law i.

Since the index are the same, the bases must be the same.

Thus,

By left cancellation of , . Add 1 to both sides.

(8) Find such that .

Solution: Adjustment.

By equating both sides, .

(9) If , where ; find P and Q.

Solution: in standard form. By collecting like bases (terms),

By equating both sides, we get:

(10) Solve .

Solution: By the law ii, we have:

Factor out common terms.

Carry out cancellations.

(11) Simplify .

Solution: This is equal to .

Since , the answer is after the cancellations and simplifications, whence .

This is very straight forward.

(12) Simplify

Solution: Since , . This is . This is a quadratic equation in y terms. [You may use formula method]

(13) Solve for x in .

Solution: since

By taking log of both sides, we get:

Divide both sides by y.

Take antilog of both sides.

Hence,

Take siinverse of both sides.

(14) Find if .

Solution: Square both sides.

Expand the bracket.

Equate both sides.

But, . Thus, and where b = 2.

This implies that a = 3 and b = 2.

Hence, .

(15) Express x interms of y in .

Solution: By using law i and ii, we have:

Divide both sides by 8.

hence

By taking log of both sides,

Divide both sides by log 2, then add 3 to both sides.

Hence,

Use the change of base law.

Exercises 1

(1) What is

(2) What is ?

(3) What is ?

(4) Evaluate.

(5) Compare .

(6) Compare

(7) Which is larger: .

(8) .

(9) Simplify .

(10)Simplify .

(11)Evaluate .

(12)Solve .

(13)Solve

(14)Simplify.

(15)Simplify.

(16) Solve.

(17) Simplify.

(18)Simplify .

(19)Find the value of y in terms of x of:

(20)What is ?

(21)Evaluate

(22)Evaluate .

(23)Simplify.

(24)Simplify.

(25)Simplify .

(26)Evaluate.

(27)Which number is equivalent to ?

(28)Solve for x in .

(29)Find x in terms of y in .

(30)What is

(31) Evaluate

(32)What is .

(33)Simplify .

(34)Find y in terms of x given that

(35) Solve for x in

(36) What is the square root of

(37) What is the quarter of

(38) Simplify

(39) Express as .

(40)

(41)Find the solutions of .

2.0.Surds

Surds are irrational numbers. By irrational numbers, we understand numbers that cannot be expressed in form of a ratio.

This topic is connected to the earlier as , as we proved. For a = 2, 3, 5, etc; is a surd.

Unlike indices, surds have fewer laws:

i. ;

ii. ; and

iii. .

(a)Simplifications of Surds

(1) Simplify: .

Solution:

(a)

(b)

(c)

Please observe that 50 is reduced to a product of two numbers, a perfect square and a prime number.

From (a), (b) and (c);

Now look at the following:

(d)

(e)

(f)

From (d), (e), (f);

I have vacuously finished simplifications, additions and subtractions of surds.

I can now delve into multiplication and division. Before then, try the following very simple exercises.

(g)

(h)

(i)

(j)Answers[(g)(h) (i)10 (j)6]

(b)Division of Surds

Here, we will present conjugation of denominator and conjugation of numerator.

Consider.

Please observe that

(c)Rationalization of Numerator

Let . a is the numerator and b is the denominator. We want to make the numerator rational. To rationalize the numerator, we multiply the equation by the value of the numerator over the value of the numerator.

For example, , i.e. the numerator 3 is now rational.

(d)Rationalization of Denominator

We try to make the denominator rational, non-surdal value (integer number) in the numerator. This trick is widespread in many texts.

Examples

(i)

(ii)

(e)Conjugation

The conjugate of is . The later neutralizes the former and vice-versa so as to have non-surdal value (rational number).

For example,

(iii)

(iv)

(v)

(vi)

(vii)

The importance of these examples lies in the fact that if one is given

, one will obviously notice that the answer is 13335-35=1300 directly.

(f) Rationalization of Denominator by Conjugation

This is plainly combination of sub topic 1.6 and 1.5.

Examples

(1) Rationalize the denominator of .

Solution:

In the above, multiplication by is like multiplication by1.

Now, . 2 will go into 1000, 500 times.

(2) Rationalize the numerator of .

Solution: The conjugate of is .

Hence,

Opening brackets.

After subtraction of common factors,

(3)Solve .

Solution: Let x = . Then truncating 1 at infinity yields: x =

Square both sides.

Thus,

Open brackets.

Collect like terms.

=

The second value is very insignificant.

What is the value of .

Solution: Let .

Then , i.e.

Hence,

Exercises 2

(1) Rationalize the numerator of

(2) Rationalize the numerator of ..


(4) Evaluate .

(5) Evaluate .

(6) Add .

(7) Which is larger

(8) Which is larger

(9) Which is larger ?

(10) Determine which is larger

(11) Evaluate .

(12) Evaluate .

(13) Evaluate

(14) Solve

(15) Add:.

(16) Simplify .

(17) Simplify .

(18) Solve.

(19) Given that , find x.

(20) Find x given that .


(22) What will rationalize

(23) Find a real number x such that .

(24) Is

(25) Is golden?

(26) Solve .

(27) Divide .

(28) Expand .

(29) Find all the values of x satisfying

3.0.Logarithm of Numbers

It is obvious that if then by laws of indices. The stress is when , what will be the value of a or the value of it in . This is the main aim of this topic.

Logarithms are of two categories: Logarithm of numbers greater than 1 and logarithm of numbers less than one. We will look at the former first and the later follows immediately.

Axioms of Logarithm

i.

ii.

iii. Zero index law

iv.

v. Negative index law

vi. . This is referred to as change of base axiom.

Remarks: In place of , you can write any arbitrary number. Please observe that log is always in small first letter and the base is smaller in size than the power.

Log means Laperian logarithm or Natural logarithm with base e, written some times as ln.

Examples

(1) Solve .

Solution: This is straight forward since .

Hence,

=18 by axiom (iv).

(2) Solve

Solution:

By law of log,

2 power 5 is 32.

(3) If , find a.

Solution: Take log of both sides.

Divide both sides by log 3.

Use change of base.

Use calculator.

(4) Find .

Solution:We know that

Use property of log.

Since

(5) Solve .

Solution:

We can observe that using change of base.

(6) Evaluate .

Solution: Here, the base are the same, and hence, whence .

The answer is after useful cancellations.

(7) Given that

(i) log 2.8 (ii) log 35 and (iii) m in

Solution:

(i)

By law (i) and (ii), it equals

which implies

That is,

(ii)

(iii)For take log of both sides.

Thus, we have: substituting from above.

(8) Solve .

Solution: This is equal to

By law of log,

Take cube root of both sides.

Take inverse of both sides.

(9) Find a given that

Solution: This is

Beware that cancellation of logs from each side are for same base.

That means which implies

(10)Find c in terms of a, b, c, d and e given that

Solution: whence .

by change of base.

(11) Find b in terms of a, c and d given that

This is a simultaneous equations and it can be tackled using our elementary knowledge of solutions of simultaneous equations.

By elimination method, Substituting the value, Thus .

Hence, .

(12) What is the value of a in

Solution: This is a quadratic equation in log a.

(b)Logarithm of Numbers using Table

In the olden days when calculators were not very common, logarithm using table was of paramount importance. It is as importance as calculators of nowadays.

I have also noticed that the use of logarithm table is time consuming, especially when tedious calculations are to be made.

In view of this, I will teach you how to solve these logarithms by using calculator without the examiners awareness. This is my own procedure, mind you, you will score all the marks and it is less time consuming.

You should also note that knowledge of the logarithm of numbers using the known procedure is also needed and very important in this method.

Remarks: Before you start solving logarithm, punch your calculator to get the answer. Then, you solve it using logarithm and confirm the answer. The negative sign belongs to only the mantissa. Again,

(c)Working with Bars

Examples

(a)

(b)

(c)

(d)

(e)

Examples

(1) Find [without use of calculator]

Solution:

Number

Logarithm

0.63954

1.8058

0.003

3.4771

213.8

2.3287

Procedure:

In 0.63954, . Shift clockwise once. Hence, this is 1 place. In 0.003, the point moves 3 places clockwise. Thus, this is 3. Punch log . That is 1+0.8059=1.8059.

Punch 0.3287 antilog. This is equal to 2.1316. Then shift the point clockwise 2 decimal places from the initial decimal position.

Now, to find antilog of 3.1214, punch antilog of 0.1214= 1.3225.

Move the point anticlockwise 3 places. That is, Note that as it is written in the second step.

(2) Find . [ without using of calculator]

Solution:

Number

Logarithm

0.0062

0.1837

(3) Find . [ without the use of calculator]

Solution:

Number

Logarithm

0.3641

66.1602

1.8206

(4) Find . [ without the use of calculator]

Solution:

Number

Logarithm

2.3127

0.3641

=1.8206

=1.8206

=

368.553

2.5665

=2.5665

(5) Find .[ without use of calculator]

Solution: By using calculator directly, we have:

We now use log table to demonstrate this.

Number

Logarithm

=0.8974 +

=0.8243

1.7217 __

0.00

=

490950000

9.8186

Exercises 3

(1) Evaluate .

(2) Evaluate .

(3) Solve .

(4) Solve

(5) Evaluate

(6) Evaluate .

(7) Evaluate

(8) Express in terms of log x, log y, log a, and log b.

(9) Find a given that .

(10) Find a given that .

(11) Solve .

(12) Simplify

(13) Solve for x in =2.

(14) Solve

(15) Trace the error: 10>8. Multiply both sides by

(16) Solve

(17) Solve

(18) Show that .

(19) Change

(20) Write in terms of .

(21) What is ?

(22) Show that

(23) Use log table to evaluate

(24) Use log table to evaluate .







(31) What is ?

(32) How can ?

(33) Find c in

(34) Solve for x in )))=1

(35) Solve this system of equations using logarithm in base ten

(36)Trace the error.

4.0.Variation

Mathematical model is formulae. In all mathematical models, some parameters or variables vary with others. In all mathematical models, there is variation. For example, in

; there is a partial and joint variations.

(a)Types of Variation

(1) Direct;

(2) Inverse;

(3)Joint; and

(4)Partial.

We will look at them one after the other.

(1) Direct Variation

This is when the term varies directly is used in a text or in a statement. For example, a varies directly as b means a varies as b. This is denoted by . The symbol is equal to =k, where k is constant of proportionality.

You can use any letter for the constant. For examples,

(i) If m varies as n, find the formula connecting m and n.

Solution:

m varies as n. When . Find the formula connecting m and n.

Solution: . By substituting for m and n, we get: implies k=

Hence, m=n is the required result.

(ii) Let a squared varies as 5th root of b. When a is 1, b is 32. Find the formulae connecting a and b. Find a when b is 16807.

Solution: , where k is constant.

For , .

The formula connecting a and b is .

Thus, . That is, by substituting for b,

(5) Given that m varies as nth root of p. Find the formula connecting m, n and p given that m=6 when n=6 and p=729 given that m=6 and n=6.

Solution: . This implies that. Thus k=2 and .

(2)Inverse Variation

This is when a parameter or variable varies inversely as another one.

Examples

(1) If ln a varies inversely as , then the formula connecting a and b given the constant of proportionality as is . Thus .

(2) Let a varies inversely as twice the cube root of b. When a=2, b=27. Find the formula connecting a and b.

Solution: which implies . Hence . The formula is .

(3)Join Variation

This is the join statement of direct and indirect variations. For example, a simpler example is when a varies as b and inversely as c. This is written as meaning . This is better viewed as . That is, , where k=constant.

(3) Let varies directly as sin b and inversely as cosec c. When Find the formula connecting a, b and c.

Solution: .

This implies. And, k=1.

The formula is

(4)Given that m varies as n power p and inversely as q power r. Find the formula connecting m, n, p, q and r. And, when m=3, .

Solution: . This implies.

By making useful substitutions, we get: k=4.

Hence the formula is .

(4)Partial Variation

This is when a variable varies partly as another variable and partly as another different or the same variable. The variable, though, may partly varies as a constant. Here, there is always need of introducing two or more constants.

Note that if either of the constants is zero, partial variation turns out to be join variation. The solution of the non-zero constant is a simultaneous equations problem.

Examples

(5) If s varies partly as product of u and t and partly as a product of square of t and a; find the formula connecting s, u, t and a given that , when a=1, u=1 and t=1. And, s=0.5 when u=1, t=1 and a=-1.

Solution: . Let the constants be .

Hence, we have: (1)

Substituting for s, u and t, we have: and. That is, . By substituting the values of into (1), we get: . This is a quadratic mathematical model for the rectilinear motions.

(6) If a varies partly as a constant and partly varies as b; when a=7, b=3 and when b=4, a=9; find a when b=5.

Solution: . That is, where are constants of proportionality.

Now, we substitute for the variables.

Thus,

A = B C

That is, A=BX, . This is the inverse matrix method.

Set .

=

Cofactor of B==.

To find a when b=5, we use:

I will come to inverse matrix method. You can use any other method of solutions of simultaneous equations for now.

Exercises 4

(1) Let the voltage V of a bulb varies directly as the sum of the resultant resistance R and internal resistance r. Find the formula connecting V, R and r.

(2) In a moving train, the acceleration a of the train varies directly as a product of the sum of its initial velocity u and its final velocity v and of its period t. Given that at initial stage, when a=1 m/s2, v=1m/s, u=1m/s and t=1s. Find the relationship between a, v, u and t.

(3) In a simple pendulum experiment, the period of revolution T varies directly as square root of its length and inversely as square root of its acceleration due to gravity. Given that g is when T=1s and l is 1m. Find the relationship btw T, l and g.

(4) The general solution of a standard form of quadratic equation when in terms of x is given by the x variesdirectly as b and inversely as a partly and varies partly as root of the difference of and 4ac and inversely as a. Given the constant of the proportionality as Solve

(5) The area A of trapezium varies directly as the product of the sum of its sides b1, b2 and its height h. When A=1, b1=1, b2=1 and h=1. Find the formula connecting A, b1, b2 and h.

(6) In an equation of motion , when a=12, t=3 and when a=10, t=2 1/2 . Find u and v.

(7) In the area of a trapezium, A=12, h=3 and when A=10, t=2 1/2. Find u and v.

(8) The Newton Law of gravitation states that the force of attraction of two bodies in space varies directly as the product of their masses and inversely as the square of their distance. Write the formula connecting f, m1, m2 and r.

(9) The potential energy Ep of a falling body varies directly as the product of its mass and the height it falls through. Given that when Ep=987joules(j), m=10kg and h=10m. Find h when m=3kg and Ep=59.22j.

(10) The kinetic energy Ek of a moving object varies directly as the product of its mass and square of its velocity when Ek =4 joules, m=2kg and v=2m/s. Find the formula connecting Ek, m and v.

(11) A varies directly as b partly and varies directly as c partly. When a=2, b=1 and c=1, and when a=3, b=1 and c=1. Find a when b=2 and c=3 and b when a=6 and c=4.

(12) The inverse of a focal length (f) of a concave mirror varies inversely as its object distance u from the mirror partly and partly varies inversely as its image distance V. When f=2cm, U=4m and V=4m, and when f=1/2 cm, u=1m and V=1m. Find the formula connecting f, u and v.

(13) The cube of c varies directly as root of a and varies inversely as root of twice b. Given the constant of proportionality as 3. Find the formula connecting a, b and c.

(14) A varies directly as square root of b partly and varies as c squared partly. Given the constant of proportionality as 2 and 3 respectively. Find a, when b=16 and c=2.

(15) C varies directly as partly and partly varies as . When c=2, . When c=05, Get the identity.

(16) If a varies directly as square root of c and b varies inversely as d, and a+b=e, then e varies as what?

(17) G varies directly as e partly and partly varies directly as f. When g=3, e=1 and f=2, when g=5, e=2 and f=4. E varies directly as root of h and f varies directly as root of square of i. Then g varies as what?

(18) Given that log p varies directly as When p=10, q=1. Find the formula connecting p and q. When q=3, find p.

(19) Sinh a varies as b. Find b when a=5 and the constant of proportionality is 9.

5.0.Algebra

We present classical Algebra. This is a technique of using symbols to represent numbers. For example, let N represent x in N 20- N 2 for 20x-2x. The answer to this is N 18 obviously because I used N, this represent 18x. Let x be a pencil and y be a biro. Then 20x+15y represents 20 pencils and 15 biros.

(a)Additions and Subtractions

Examples:

(1)

(2)

(3)

(4)

(5)

(6)

We are almost through with additions and subtractions of Algebra. Next, we now demonstrate multiplication and division of it.

(b)Multiplication

Examples

A. (7)

(8)

(9)

B. (10)

(11)

(12)

(13)

You should note that factorization is the antonym of expansion.

C. (14)

(15)

(16)

(17)

(18)

The last two examples are trying to lead us to two different subtopics: Difference of Two Squares and the Pascal Triangle.

(c)Division

Examples

(19) if

(20)

(21)

(22)

(23). This referred to as Decomposition. We will use it throughout this text.

(24)

(d)Difference of Two Squares

This is subtraction of square of a variable from a square of another different variable. For example, . For this to be vivid, we look at the converse. This is a matter of demonstration only since 24-4=21 directly.

Examples

A.

(25)

(26)

(27)

(28)

B.

(29) Solve .

Solution: Use difference of two squares on the given question to get:

(30) Solve .

Solution:

(31) Factorize .

Solution:

(e)Pascal Triangle

This is a triangle showcasing coefficients of , where . The triangle is as follows:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

Remarks: We are only considering the coefficients for example 121 is from , 1331 is from , etc. as the index of a is decresing, the index of b is increasing and vice versa. This is an application of combination. It is precisely .

(f)Zero Fractions and Undefined Fractions

When . When , . That is true for every a, where a is a real number. The result is true for even.

Examples

(32) For what value of x is

(a) , zero; and (b)

Solution:

(a) . This implies. This implies x=1.

(b) implies x=1. The value is 1.

(33) For what values of x and y is undefined?

Solution: .

For the above to be undefined, . That is the following set of solutions:(1,1), (2, 2),

(-3, -3), (-1, -1), etc.

(34) For what values of x is zero?

Solution: . This implies . This implies .

Exercises 5

(1) If

(2) Given that

(3) What is

(4) What is

(5) What is the meaning of 9 month ago?

(6) What is my age in a decade to come.

(7) If 1 less twice a square of a certain number is a square of the number. What is the number?

(8) When is undefined?

(9) When is zero?


(11) When is question 10 equals to zero?

(12) Solve 25-4.

(13) Evaluate

(14) Evaluate .

(15) What is the diagonal sequence of the Pascal Triangle?

(16) When is

(17) What are the factors of -2?

(18) Divide

(19) Divide


(21) When is equation in 20 equals to zero?


(23) When is

(24) Evaluate

(25) Evaluate


(27) When is zero?

(28) What is ?

(29) Expand


(31) Find x in .

(32) For what values of x is undefined?

(33) For what values of x is .



(36) What is for

6.0.Subject Formula

In English, subject is the noun or pronoun or group of words making such function. A sentence is talking about its subject you know. In mathematics too, formula is a sentence, where is the verb. Like in English, the subject should be followed by the verb especially in declarative sentences. Hence in mathematics too, the variable to be made the subject should come before .

If , then a is not the subject for a appears in either sides of the equality.

Examples

(1) Make a the subject in

Solution: By adding to both sides, we get: .This is what is required.

(2) Make a the subject in .

Solution: . By cross multiplication, we have: . This implies


Solution: By taking square root of both sides, we have: .

This implies.

(4) Make a the subject if .

Solution: By taking cube of both sides, we have: . This implies .

(5) Make a the subject if

Solution: By adding to both sides; we have .

By factoring a, we get:

Hence,

(6) Make f the subject if

Solution: By adding to both sides, .

By taking LCM, , i.e, . Thus .

(7) Make g the subject in . Hence find g when .

Solution: Square both sides.

By cross multiplication, we have: .

(8) Make the subject in .

Solution:. This implies . Thus .


Solution:

Open bracket.

This is a quadratic equation.

(10) Make the subject in .

Solution:u

Divide both sides by

This is a quadratic equation with a=1, b=


Solution:

In standard form, we have:

This is a quadratic equation that you can easily solve using formula method.

(12) Make n the subject in

Solution:

In standard form of quadratic equation, we have:

This is a quadratic equation having , .

(13) Make r the subject in .

Solution: To make r the subject, you need to introduce logarithm noticing that r2 and 2

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