Real Numbers and Their Operations

[Pages:28]Algebra 1

Math Study Guides

Real Numbers and Their Operations

Real Numbers and the Number Line

Natural Numbers ? The set of counting numbers { 1, 2, 3, 4, 5, ...}. Whole Numbers ? Natural numbers combined with zero { 0, 1, 2, 3, 4, 5, ...}. Integers ? Positive and negative whole numbers including zero {...,-5, -4, -3,-2, -1, 0, 1, 2, 3, 4, 5...}. Rational Numbers ? Any number of the form a/b where a and b are integers where b is not equal to zero. Irrational Numbers ? Numbers that cannot be written as a ratio of two integers.

When comparing real numbers, the larger number will always lie to the right of smaller numbers on a number line. It is clear that 15 is greater than 5, but it might not be so clear to see that -5 is greater than -15.

Use inequalities to express order relationships between numbers. < "less than" > "greater than" "less than or equal to" "greater than or equal to"

It is easy to confuse the inequalities with larger negative values. For example, -120 < -10 "Negative 120 is less than negative 10."

Since -120 lies further left on the number line, that number is less than -10. Similarly, zero is greater than any negative number because it lies further right on the number line.

0 > -59 "Zero is greater than negative 59."

Write the appropriate symbol, either < or >.



Page 1

Algebra 1

Math Study Guides

List three integers satisfying the given statement. (Answers may vary.)

Absolute Value ? The distance between 0 and the real number a on the number line, denoted |a|. Because the absolute value is defined to be a distance, it will always be positive. It is worth noting that |0| = 0.



Page 2

Algebra 1

Math Study Guides

Point of confusion: You may encounter negative absolute values like this -|3|. Notice that the negative is in front of the absolute value. Work the absolute value first, then consider the opposite of the result. For example,

-|3| = -3 -|-7| = -7

Believe it or not, the above are correct! Look out for this type of question on an exam.



Page 3

Algebra 1

Math Study Guides

Adding and Subtracting Integers

When first learning how to subtract real numbers, it is useful to think of this operation as adding opposite numbers. For example, 5 - 2 can be thought of as adding five with the opposite of two 5 + (-2). This may seem cumbersome, but sometimes it can help in understanding problems where you wish to subtract a negative number.

Two sequential negative signs, as seen above, is equivalent to addition. When adding two negative numbers the result will be negative. For example, if you spend $12.00 for a pair of shoes and $25.00 for a bag then you will owe $37.00 at checkout, -12 - 25 = -37.

Add and Subtract.



Page 4

Algebra 1

Math Study Guides



Page 5

Algebra 1

Math Study Guides

Multiplying and Dividing Integers

Multiplying a negative number by a positive number will result in a negative number. And when multiplying two negative numbers the result will be positive. Be careful when simplifying 5(-6), the operation here is multiplication NOT subtraction so 5(-6) = -30. The rules for division are the same.

(Negative) x (Positive) = (Negative) Example: (-7)(+4) = -28

Also, zero times anything is zero.

(Negative) x (Negative) = (Positive) Example: (-7)(-4) = +28

Multiply or Divide.

Dividing by zero is undefined. What happens when you try to divide by zero on a calculator? Zero and Division



Page 6

Algebra 1

Divide.

Math Study Guides



Page 7

Algebra 1

Math Study Guides

Fractions

Fractions can be a barrier to beginning algebra students. Also referred to as rational numbers, fractions are simply real numbers that can be written as a quotient, or ratio, of two integers.

Equivalent fractions can be expressed with different numerators and denominators. For example,

If you eat 4 out of 8 slices of pizza, that is the same as eating one-half of the pie. Usually we will be required to reduce fractions to lowest terms. Fractions in lowest terms have no common factors in the numerator and denominator other than 1.

An alternative and more common method of reducing is to identify the greatest common factor (GCF) of the numerator and denominator and then divide both by that number.

An improper fraction is one where the numerator is larger than the denominator. To convert an improper fraction to a mixed number, simply divide. The quotient is the whole number part and the remainder is the new numerator.



Page 8

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download