Expressions Using Letter-Numbers

Two friends, Shabnam and Aftab, have ages that are always 3 years apart: Shabnam is the older one. When Aftab was 10, Shabnam was 13. When Aftab turns 18, Shabnam will be 21. The relationship never changes even though both ages keep changing every year. Writing this out in words every time is tiring, so mathematicians agree on a shortcut: pick a letter to stand for Aftab's age, say a, and a letter for Shabnam's age, say s. The relationship becomes s = a + 3. Letters used this way to stand for numbers are called letter-numbers, and an expression built from them, like a + 3, is called an algebraic expression. The letter itself is not magic: it is just a nickname for whatever number happens to be true at the time. Replace a with 18 and the expression tells you Shabnam's age instantly: 18 + 3 = 21.

Before letter-numbers arrive, arithmetic expressions like 23 - 10 x 2 or 68 - (18 + 13) already have rules for finding their value: write the expression as a sum of terms, then swap and group the terms in whatever order is easiest, since swapping and grouping never change the total. A bracket with a minus sign in front, like -(18 + 13), opens up to -18 + -13: the minus sign applies to every term inside. None of these rules are new inventions for algebra. They are exactly the same rules used for plain numbers, just now applied to expressions that also contain letter-numbers. That is the entire reason algebra is trustworthy: a letter-number is standing in for some ordinary number, so anything that is always true for ordinary numbers is automatically true for the letter-number as well, whatever value it eventually turns out to have.

Look at the sequence 4, 8, 12, 16, 20: the multiples of 4. The 3rd term is 4 x 3, the 29th term is 4 x 29, and in general the nth term is 4 x n. Writing the multiplication sign every single time gets cluttered, so algebra adopts a shorthand: 4 x n is written 4n, the number first and the letter immediately after, with no sign at all between them. This is purely a writing convention, nothing more. 4n still means exactly 4 x n. To find the value of an expression like 7k when k = 4, replace k with 4 and multiply as usual: 7 x 4 = 28. To find the value of 5m + 3 when m = 2, remember 5m means 5 x m, so the expression becomes 5 x 2 + 3 = 13.

A shop sells pencils at a price of c rupees each, and on three different days it sells 5, 3 and 10 pencils. The total money earned from pencils is 5c + 3c + 10c. This means c is added to itself 5 times, then 3 times, then 10 times, which is the same as adding c to itself a total of 5 + 3 + 10 = 18 times: that is exactly the distributive property, used in reverse. So 5c + 3c + 10c simplifies to 18c. Terms that share the same letter-number, like 5c, 3c and 10c, are called like terms, and like terms can always be combined this way. If the same shop also sells erasers at d rupees each, the total earned from erasers might simplify to 11d, and the grand total becomes 18c + 11d. This CANNOT be simplified any further, because 18c and 11d involve different letter-numbers: they are unlike terms, standing for two different unknown quantities that need not be related to each other at all.

Split a big rectangle into two smaller rectangles side by side, both sharing the same height v, with widths 4 and 3. The area of the whole big rectangle can be worked out two different ways. Adding up the two smaller pieces gives 4v + 3v. Measuring the big rectangle directly, its width is 4 + 3 = 7 and its height is v, so its area is 7v. Both calculations describe the exact same rectangle, so they must give the same answer: 4v + 3v = 7v. This is not a coincidence to memorise, it is a fact that is guaranteed to be true, because both expressions are just two different, equally valid ways of measuring one real area. Whenever two expressions always take the same value no matter what number the letter-number is replaced by, they are called equal, and one is called the simplified form of the other.

Are 5u and 5 + u the same expression? They certainly look similar, both involving a 5 and a u. But 5u means 5 times u, while 5 + u means 5 more than u: these are two completely different operations, and there is no reason to expect them to give the same answer. Check with u = 2: 5u is 5 x 2 = 10, while 5 + u is 5 + 2 = 7. Already different. Check with u = 11: 5u is 55, while 5 + u is 16. Still different, and by a much bigger gap now. The two expressions are not equal, and never will be equal for any value of u except one very special value (try to find it). The lesson is that looking similar on paper is not proof of being equal: only checking that two expressions genuinely always agree, for every value of the letter-number, or better still explaining algebraically why they must agree, counts as proof.

Mark any 2 by 2 square of dates on a calendar page. Add the two numbers on one diagonal, then add the two numbers on the other diagonal: the two sums always come out equal. Checking this on five or six different squares makes it look true, but a calendar page only has a limited number of squares, so checking every single one by hand is impossible, and checking a few is not a proof. Algebra can finish the job completely. Call the top-left date of any such square a. Because each date is one more than the date to its left, the square looks like a, a+1 on top and a+7, a+8 on the bottom, since the next row down always adds 7. One diagonal sum is a + (a + 8) = 2a + 8. The other diagonal sum is (a + 1) + (a + 7) = 2a + 8. The two sums are both 2a + 8, whatever a happens to be, so they are guaranteed to be equal for every possible 2 by 2 square that could ever exist on any calendar, past or future. This is the real power of algebraic expressions: instead of checking cases one at a time forever, a single argument with a letter-number covers every case at once.