English Tutorial #01: Which and That


Which vs. That in Restrictive and Non-restrictive Clauses

If you’re puzzled about whether to use which or that, don’t panic!
It’s really not too hard if you remember these two types of clause: Nonrestrictive Clauses and Restrictive Clauses.

A nonrestrictive clause can be eliminated from a sentence without losing the basic meaning of the sentence. We should use the word ‘which’ for these types of clauses. Non-restrictive clauses will be set apart from the rest of the sentence by dashes, commas, or parentheses as in these three examples:

 

  1. The watch–which had been in her family for over sixty years–was stolen.
  2. Jamie’s kite, which was blue and green, was flying high in the sky.
  3. Tiffany’s graduation speech (which had caused a great deal of controversy at the commencement ceremony) was printed in the local paper.

 

Try reading the sentences without the non-restrictive clauses.
Is the basic message of each sentence the same?

A restrictive clause is a clause that is vital to the meaning of the sentence. Don’t even try to take these clauses out of the sentence unless you really want to change the meaning! We use the word ‘that’ to mark these types of clauses.

Because restrictive clauses are vital to sentences, they will not be set apart with punctuation as shown in these examples. After you have read the sentences, go back and circle the clauses (the part that begins with ‘that’). If you read what’s left of the sentences, you can see that much of the meaning has been lost or changed. For Example:

  1. The dog that is running down the street is mine.
  2. The watch that my mother bought me for my birthday is my favorite watch!
  3. The tests that I have studied for usually go really well.

We learnt something new today…pretty cool, eh?

Exercises:

Now it’s your chance to practice the new grammar knowledge you’ve gained!
Fill in the blanks with either which or that depending on the sentence. Good luck!

1. Down by the mall there is a store ______reminds me of the store by Missy’s house.

2. We walked through the entire store ( ______ sold all kinds of kid’s toys) just to see Santa.

3. She saw the movie ______ had recently been released with a group of her friends.

4. India is a country ______ I would like to visit someday.

5. Frankenstein, _______ was written by Mary Shelley when she was only seventeen, touches upon many issues that are relevant in our modern society.

6. When we were driving to John’s house, we saw the accident _____ had held up traffic on the freeway.

7. I jogged every morning around the park ______ had the goat statue.

8. Maggie worried that she had forgotten to pay her phone bill _____ was due yesterday.

9. Bobby and Ryan were raising money for the new equipment, _______ included new balls and bats as well as new uniforms.

10. We awaited the arrival of the books _____ we had ordered online.

 

Answers:

1. that, 2. which, 3. that, 4. that, 5. which, 6. that, 7. that, 8. that, 9. which, 10 that.

Mathematics Tutorial #03: Number Systems

Article (C): Progressions 1 – Arithmetic Progressions


Imagine that you need to build a flight of stairs out of wooden steps. Each step has a height of 0.5 m. How many steps would you need to lay in order to gain a height of 10 m?
That’s quite easy. We simply divide the required height (10 m) by the height of each step (0.5 m), which gives us the answer 20 steps.

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This is an example of Arithmetic Progression. Each step has a fixed height and laying one step on top of the other adds to the height by a constant difference.

Now let us consider sequences of numbers: Consider these two common sequences “1, 3, 5, 7…” and “0, 10, 20, 30, 40… “

It is easy to see how these sequences are formed. They each start with a particular first term, and then to get successive terms we just add a fixed value to the previous term. In the first sequence we add 2 to get the next term, and in the second sequence we add 10.

So the difference between consecutive terms in each sequence is a constant.


Note:

We could also subtract a constant instead, because that is just the same as adding a negative constant. For example, in the sequence 8, 5, 2, −1, −4… the difference between consecutive terms is −3.


 

Any sequence with this property is called an arithmetic progression or AP for short.

We can use algebraic notation to represent an arithmetic progression.

Let ‘a’ be the first term of the sequence, and let ‘d’ be the common difference between successive terms.

For example, our first sequence could be written as 1, 3, 5, 7, 9, . . . 1, 1 + 2, 1 + 2 × 2, 1 + 3 × 2, 1 + 4 × 2, . . . , and this can be written as a, a + d, a + 2d, a + 3d, a + 4d, . . . where a = 1 is the first term, and d = 2 is the common difference.

If we wanted to write down the n-th term, we would have Tn = a + (n − 1) d, because if there are ‘n’ terms in the sequence there must be (n − 1) common differences between successive terms, so that we must add on (n − 1) d to the starting value a.

We also sometimes write ‘ℓ’ for the last term of a finite sequence, and so in this case we would have ℓ = a + (n − 1) d.

Another Example:

We draw a triangle and from the mid points of the sides of each triangle, we construct another.
We continue doing this for ‘n’ times. At each iteration, the total number of triangles increases by 3, since the new triangle divides the old one into 4 smaller triangles.

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The sum of an arithmetic series

Sometimes we want to add the terms of a sequence. What would we get if we wanted to add the first ‘n’ terms of an arithmetic progression?

We would get Sn = a + (a + d) + (a + 2d) + . . . + (ℓ − 2d) + (ℓ − d) + ℓ. Now this is now a series, as we have added together the ‘n’ terms of a sequence.

This is an arithmetic series, and we can find its sum by using a trick. Let us write the series down again, but this time we shall write it down with the terms in reverse order.

We get Sn = ℓ + (ℓ − d) + (ℓ − 2d) + . . . + (a + 2d) + (a + d) + a.

We are now going to add these two series together.

We get 2Sn = (a + ℓ) + (a + ℓ) + (a + ℓ) + . . . + (a + ℓ) + (a + ℓ) + (a + ℓ), and on the right-hand side there are ‘n’ copies of (a + ℓ).

So we get 2Sn = n (a + ℓ).
=> Sn = {n (a + ℓ)} /2.

We have found the sum of an arithmetic progression in terms of its first and last terms, ‘a’ and ‘ℓ’, and the number of terms ‘n’.

We can also find an expression for the sum in terms of ‘a’, ‘n’ and the common difference ‘d’.

To do this, we just substitute our formula for ‘ℓ’ into our formula for Sn.

From ℓ = a + (n − 1) d, and
Sn = {n (a + ℓ)} /2, we obtain

Sn = {n (a + a + (n − 1) d)} /2

= {n (2a + (n − 1) d)} /2.

To sum up, we have Sn = {n (a + ℓ)} /2

                                     = {n (2a + (n − 1) d)} /2

Now here’s one to think about: A certain arithmetic sequence consists entirely of positive integers. For this sequence, the first term a1 = 1 and the number 2008 appears somewhere in the sequence. How many different arithmetic sequences have these properties?

Ans:- Let the n-th term of the sequence be 2008.
Therefore, an = a1 + (n-1) d, where ‘d’ is the common difference.
or, 2008 = 1 + (n-1) d
or, 2007 = (n-1) d
i.e., ‘d’ is a divisor of 2007. Since the divisors of 2007 are 1, 3, 9, 223, 669, and 2007, there are 6 possible values of ‘d’. Hence, the answer is that there are 6 such sequences.


Exercises:

  1. If ‘p’ times the ‘p’-th term of an AP is equal to ‘q’ times the ‘q’-th term, prove that its (p + q)-th term is zero, provided p ≠ q.
  2. The first term of an AP is – 2 and 11th term is 18. Find its 15th term.
  3. The 12th term of an AP is – 28 and 18th term is – 46. Find its first term and common difference.
  4. Which term of the AP 5, 2, –1, …. is – 22?
  5. If the p-th, q-th and r-th terms of an AP are x, y and z respectively,
    prove that: x (q – r) + y (r – p) + z (p – q) = 0
  6. Find the following sum 2 + 5 + 8 + 11 + …. + 59
  7. Find the sum of all natural numbers between 1 and 1000 which are divisible by 7.
  8. The sum of first three terms of an AP is 36 and their product is 1620. Find the AP
  9. The sum of any three consecutive terms of an AP is 21 and their product is 231. Find the three terms of the AP.

Mathematics Tutorial #02: The Number System

 

Article (B): An Introduction to number systems.


Since times immemorial, man has felt the need to quantify everything around him. This need is what drove us to count objects and everyday things. The earliest number systems to be used were the tally system, the hieroglyphic system, the cuneiform system, the Chinese, the Greek, the Roman, the Mayan and the Devanagari systems.
Eventually, number systems evolved to include several new concepts. The number systems used most frequently now are the modern number systems.

The Modern Number Systems:


 

Natural numbers  – Start from 1 and is the set of all counting numbers {1, 2, 3, …}

Whole Numbers – Starts from 0 and includes the set of natural numbers.

Integers  – Positive and negative counting numbers, as well as 0: {… , -2, -1, 0, 1, 2, …}

Rational numbers
 – The set of all numbers that can be represented as a fraction. All integers are rational, but the converse is not true.

Irrational numbers – The set of all numbers that are not rational, i.e., cannot be expressed as a fraction.

Real numbers
 – The set of all numbers counting, rational, irrational, transcendental and two elements – ∞ and + ∞ together make up the real number system.

Complex numbers – A complex number is a number that can be expressed in the form a + bi, where ‘a’ and ‘b’ are real numbers and ‘i’ is the imaginary unit, that satisfies the equation:-
x^2= -1, that is, i^2= -1.
In this expression, ‘a’ is the real part and ‘b’ is the imaginary part of the complex number.


Note:

1. We will come back to study Complex numbers later on in greater detail.
2. Transcendental numbers are real or complex numbers that are not algebraic—that is, they are not the roots of any non-zero polynomial equation with rational coefficients. The most prominent transcendental numbers are π and e.


 

 

Mathematics Tutorial #01: The Number System

 

Article (A): Infinity and related concepts


What actually is infinity?
Often come across the term infinity (∞) in mathematics. For example: tan 90 = ∞

This brings us to a very important question. What does infinity mean? Infinity is not a real number. It is the concept of an unimaginably large number.

Take the following example:

Let x = a/b, where ‘a’ and ‘b’ are real numbers. Now, as ‘b’ grows smaller and smaller, the value of ‘x’ becomes larger and larger. Let us consider the value of ‘x’ as ‘b’ tends towards 0. The value of ‘x’ tends towards an infinitely large number we term as infinity. What we usually term to be ‘infinity’ is actually either ‘undefined’ or ‘indeterminate’.

 

Why division by zero is not allowed: The difference between undefined and indeterminate.


 

Let us assume for the sake of argument that there exist two real numbers ‘a’ and ‘b’ such that:

a/0 = b, a ≠ 0.
=> a = 0.b
=> a = 0 for any real number b, which is a contradiction.

Hence, the form a/0 is ‘undefined’ when a≠ 0.

Now, we assume that a = 0.
Then this implies that b.0 = 0, which holds identically for any real ‘b’ (1.0 = 0, 2.0 = 0 … and so on).

As a result, the value of ‘b’ cannot be determined. And thus, division by zero is not allowed.
As such, the form “x = 0/0” is ‘indeterminate.’

A few more examples:

Division by zero is an operation for which you cannot find an answer, so it is disallowed. You can understand why if you think about how division and multiplication are related.

12 divided by 6 is 2   because 6 times 2 is 12
12 divided by 0 is x   would mean that 0 times x = 12

But no value would work for ‘x’ because 0 times any number is 0. So division by zero doesn’t work.

Another way to look at it is to imagine that there is a magic coin that takes up no space at all. How many such coins can you put into your coin box? Some people may say that the answer is “infinite”. But the real answer is – THERE IS NO ANSWER!