numbers?
A. All the numbers except
Q. Given the numbers: 8, 11, 21, 51, 67, which can be classified as prime and which are composite?
A. The numbers 11 and 67 are prime. Their only divisors are 1 and themselves. The number 8 is composite, because it can be written as 2 · 4. The number 21 is composite, because it can be written as 3 · 7. And the number 51 can be written as 3 · 17, so it’s composite.
Practice Questions
1. Determine which of the classifications correspond to the numbers.
2. Determine which of the classifications correspond to the numbers.
Practice Answers
1.
2.
Algebra and symbols in algebra are like a foreign language. They all mean something and can be translated back and forth as needed. It’s important to know the vocabulary in a foreign language; it’s just as important in algebra.
✓ An expression is any combination of values and operations that can be used to show how things belong together and compare to one another. 2x2 + 4x is an example of an expression. You see distributions over expressions in Chapter 9.
✓ A term, such as 4xy, is a grouping together of one or more factors (variables and/or numbers) all connected by multiplication or division. In this case, multiplication is the only thing connecting the number with the variables. Addition and subtraction, on the other hand, separate terms from one another. For example, the expression 3xy + 5x – 6 has three terms.
✓ An equation uses a sign to show a relationship – that two things are equal. By using an equation, tough problems can be reduced to easier problems and simpler answers. An example of an equation is 2x2 + 4x = 7. See Chapters 14 through 18 for more information on equations.
✓ An operation is an action performed upon one or two numbers to produce a resulting number. Operations are addition, subtraction, multiplication, division, square roots, and so on. See Chapter 7 for more on operations.
✓ A variable is a letter representing some unknown; a variable always represents a number, but it varies until it’s written in an equation or inequality. (An inequality is a comparison of two values. For more on inequalities, turn to Chapter 19.) Then the fate of the variable is set – it can be solved for, and its value becomes the solution of the equation. By convention, mathematicians usually assign letters at the end of the alphabet to be variables to be solved for in a problem (such as x, y, and z).
✓ A constant is a value or number that never changes in an equation – it’s constantly the same. Five is a constant because it is what it is. A variable can be a constant if it is assigned a definite value. Usually, a variable representing a constant is one of the first letters in the alphabet. In the equation ax2 + bx + c = 0, a, b, and c are constants and the x is the variable. The value of x depends on what a, b, and c are assigned to be.
✓ An exponent is a small number written slightly above and to the right of a variable or number, such as the 2 in the expression 32. It’s used to show repeated multiplication. An exponent is also called the power of the value. For more on exponents, see Chapter 5.
Practice Questions
1. How many terms are there in the expression: 4x – 3x3 + 11?
2. How many factors are found in the expression: 3xy + 2z?
3. Which are the variables and which are the constants in the expression:
4. Which are the exponents in the expression: z2 + z1/2 – z?
Practice Answers
1. 3. The term 4x is separated from 3x3 by subtraction and from 11 by addition.
2. 5. There are two terms, and each has a different number of factors. The first term, 3xy has three factors: the 3 and the x and the y are multiplied together. The second term, 2z, has two factors, the 2 and the z. So there are a total of five factors.
3. The variables are h, b1, and b2; the constant is the fraction
4. 2,
In algebra today, a variable represents the unknown. (You can see more on variables in the “Speaking in Algebra” section earlier in this chapter.) Before the use of symbols caught on, problems were written out in long, wordy expressions. Actually, using letters, signs, and operations was a huge breakthrough. First, a few operations were used, and then algebra became fully symbolic. Nowadays, you may see some words alongside the operations to explain and help you understand, like having subtitles in a movie.
By doing what early mathematicians did – letting a variable represent a value, then throwing in some operations (addition, subtraction, multiplication, and division), and then using some specific rules that have been established over the years – you have a solid, organized system for simplifying, solving, comparing, or confirming an equation. That’s what algebra is all about: That’s what algebra’s good for.
Deciphering the symbols
The basics of algebra involve symbols. Algebra uses symbols for quantities, operations, relations, or grouping. The symbols are shorthand and are much more efficient than writing out the words or meanings. But you need to know what the symbols represent, and the following list shares some of that info. The operations are covered thoroughly in Chapter 5.
✓ + means add or find the sum or more than or increased by; the result of addition is the sum. It also