Lesson 1 · from §1, 2 of the 1910 text
Symbols
Algebra deals with many topics that are new to you, yet you'll find the subject is a continuation of your work in arithmetic.
Symbols
Symbols are used far more in algebra than in arithmetic, and many new ideas come from their meaning and use. Some symbols represent numbers, others indicate operations on numbers, and others represent relations between numbers. Arabic numerals and letters are used to represent numbers.
The operation symbols +, -, \times, and \div mean the same as in arithmetic: + is read plus, - minus, \times multiplied by, and \div divided by.
The multiplication sign is usually replaced by a dot or left out entirely. For example, 3 \times a is written 3 \cdot a or 3a, and 2 \times a \times b is written 2ab. Also, a \div b is often written \dfrac{a}{b}. The sign = is read equals or is equal to.
Quick check
Write 5 \times a \times b the algebra way, without any multiplication sign.
Try it: if h is the number of seconds in an hour, then h = 3600, so 2h means 2 \times 3600 = 7200 seconds.
Quick check
If m is the number of seconds in a minute, how many seconds does 5m stand for?
From the history of algebra
Origin of symbols. Many symbols in common use today have histories that are interesting in themselves and show how slowly algebra developed. Symbols that seem meaningless often began as abbreviations long since forgotten, and methods we find hard to master sometimes took hundreds of years to perfect.
In the early centuries there were practically no algebraic symbols; people wrote out the words plus, minus, equals, and so on. In the sixteenth century several Italian mathematicians used the initial letters \bar{p} and \bar{m} for + and -. Some think our modern - came from writing the letter m so quickly that its curves flattened into a straight line, and that + may have come from a hastily written p. Others believe these symbols were first used in German warehouses in the fifteenth century to mark the weights of boxes: if a box meant to weigh 100 pounds was 5 pounds over, it was marked 100 + 5; if 5 pounds short, 100 - 5. Although the first book to use these symbols was published in 1489, they weren't in common use until about 1630.
Both multiplication symbols were first used around 1630. The cross was used by two Englishmen, Oughtred and Harriot, and the dot first appears in the writings of the Frenchman Descartes. (Harriot was sent to America in 1585 by Sir Walter Raleigh and made the first survey of Virginia and North Carolina.)
The line for division was used long before any of these. The Arabs, as early as 1000 A.D., wrote both \frac{a}{b} and a/b for the quotient of a by b. The symbol \div didn't appear until about 1630.
The word equals was usually written out until about 1600. The modern sign = was probably introduced by the Englishman Robert Recorde in 1557, because, as he put it, "Noe. 2. thynges can be moare equalle" than two parallel lines. It wasn't accepted right away, and for the next fifty years symbols like \parallel were also used.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Express 4h + 3m in seconds, if h and m stand for the number of seconds in an hour and in a minute respectively.
Express 5y + 4f in inches, if y and f stand for the number of inches in a yard and in a foot respectively.
If q and d represent the number of cents in a quarter and in a dime respectively, find the value in cents of 4q + 6d.
If t and h represent the number of pounds in one ton and in one hundredweight respectively, express 4t + 6h in pounds.
Simplify: 3x + 5x
Simplify: 4x + 5x
Simplify: 2x + 3x + 6x
Simplify: 2x + 2 + 3x + 4
Simplify: x + x + 2 + x + x + 2
Simplify: n + n + 1 + n + 2
Simplify: 5a + 18 - 3a - 7
Simplify: 8x - 3 + 18 - 5x
Simplify: 4w - 8 + 3w + 20
If m represents one month and y = 12m, express y + 5m in terms of m.
If y (one yard) equals 3f (f = one foot), express 2y + 7f in terms of f.
Express 4q + 3n in terms of n, if q (one quarter) equals 5n (n = one nickel).
Express 2d + 15h in terms of h, if d = 24h.
Express 15h + 50m in terms of m, if h = 60m.
The side of a square is 5 inches. What is its area? Its perimeter?
The side of a square is s inches. What represents its perimeter? Its area?
The base of a rectangle is 12 feet and its altitude is 4 feet. What is its area? Its perimeter?
If b represents the number of feet in the base of a rectangle and a the number of feet in its altitude, what is its area? Its perimeter?
A rectangle is twice as long as it is wide. Let x represent the number of inches in its width. Then express (a) the length in terms of x; (b) the perimeter; (c) the area.
A man is three times as old as his son. If y is the number of years in the son's age, what will represent the father's age?
A father is 30 years older than his son. If y represents the son's age in years, what will represent the father's age?
A rectangle is 12 feet longer than it is wide. Let w represent the width in feet. Then represent the length and the perimeter in terms of w and some numbers.
A rectangle is 18 feet narrower than it is long. If w represents the width in feet, what will conveniently represent the length? The perimeter?
A rectangle is 4 feet longer than twice its width. Let w be the width in feet, and express the length and the perimeter in terms of w.