Lesson 3 · from §4, 5, 6, 7, 8 of the 1910 text
Factors, Exponents, and Coefficients
Representing numbers
In algebra, numbers are represented by one or more numerals, by letters, or by both combined. For example, 3, 25, a, 2b, 4xy, and 2x + 3 are all algebraic symbols for numbers.
Exactly what numbers 4xy and 2x + 3 stand for isn't known until we know what x and y stand for. In one problem they might have very different values than in another. Working out those values in the problems that come up is the main aim of algebra.
Factors
A factor of a product is any one of the numbers that are multiplied together to form the product.
For example, 3ab means 3 times a times b, so 3, a, and b are each factors of 3ab. Similarly, 4(a + b) means 4 times the sum of a and b, so 4 and a + b are factors of 4(a + b).
Exponents
An exponent is a number written to the right of and above another number to show how many times that number is used as a factor.
3^2 = 3 \cdot 3, \qquad 5^3 = 5 \cdot 5 \cdot 5, \qquad a^4 = a \cdot a \cdot a \cdot a, \qquad 4xy^3 = 4 \cdot x \cdot y \cdot y \cdot y
In a^b, b is the exponent of a. If a is 4 and b is 3, then a^b = 4^3 = 4 \cdot 4 \cdot 4. An exponent of 1 usually isn't written. (Later this definition will be extended to include fractions and other numbers as exponents.)
Quick check
Write x \cdot x \cdot x \cdot x \cdot x using an exponent.
Coefficients
If a number is the product of two factors, either factor is called the coefficient of the other.
In 4x^2y, 4 is the coefficient of x^2y, y is the coefficient of 4x^2, and 4y is the coefficient of x^2. A coefficient of 1 is usually left out. When there's a numerical coefficient other than 1, it's usually written first: we write 5x, not x5.
These examples show the difference between a coefficient and an exponent:
3x = x + x + x, \qquad x^3 = x \cdot x \cdot x
If x = 5, then 3x stands for 15, while x^3 stands for 125. If x = 10, then 3x = 30, while x^3 = 1000.
Quick check
If x = 4, what do 3x and x^3 stand for?
Parentheses
When two or more numbers joined by operation signs are enclosed in parentheses, the whole expression is treated as a single number.
So 3(6 + 4) means 3 \cdot 10, or 30; (17 - 2) \div (8 - 3) means 15 \div 5, or 3; (5 + 7)^2 means 12^2, or 144; and 6(x + y) means six times the sum of x and y.
Quick check
What does 2(9 - 4) stand for?
Roots
As in arithmetic, the symbol for square root is \sqrt{\phantom{x}} (or \sqrt[2]{\phantom{x}}), and the symbol for cube root is \sqrt[3]{\phantom{x}}. The name radical sign applies to all symbols like \sqrt{\phantom{x}}, \sqrt[3]{\phantom{x}}, \sqrt[4]{\phantom{x}}, and so on. The small number in a radical sign, like the 3 in \sqrt[3]{\phantom{x}}, is called the index.
Typing answers: write powers with
^(so a^2 isa^2), square roots assqrt(...), and cube roots ascbrt(...). You can leave out multiplication signs:3abmeans 3 \cdot a \cdot b.
From the history of algebra
There has been a lot of variety in the symbols for roots. The symbol \sqrt{\phantom{x}} was introduced in 1544 by the German mathematician Stifel, and it's a corrupted form of the letter r, the first letter of the Latin word radix, meaning "root." Before then, square root was written with the symbol ℞, which doctors still use on prescriptions as an abbreviation of recipe. So \sqrt[4]{5} would have been written ℞⁴5. Some early writers used a dot for square root, writing \sqrt{2} as \cdot 2, and the Arabs marked a root with an Arabic letter placed directly over the number.
Exercises
Write in symbols:
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.
The sum of three times a and five times b.
Three times a subtracted from five times b.
The square of a subtracted from the square of b.
Two times a squared subtracted from three times a squared.
The quotient of a divided by b.
The product of four times a squared and b.
The sum of a and b divided by their difference.
The product of a and 2b - c.
The product of a by the sum of b and c.
The result of subtracting a - b from 7x.
The product of the sum of a and b by their difference.
The sum of the square root of 5a and the cube root of 7b.
The product of x - y and the square root of 7x.
The square of the sum of a and b.
The square of the difference of a and b.
The quotient of three times a times the square of b by four times c times the cube of a.
The sum of the quotients of a by 3x and 4y by c.