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Lesson 4C

Lesson 4H. Lesson 4D. Lesson 4A. Lesson 4E. Lesson 4I. Lesson 4B. Lesson 4J. Lesson 4F. Lesson 4G. Lesson 4C.

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Lesson 4C

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  1. Lesson 4H Lesson 4D Lesson 4A Lesson 4E Lesson 4I Lesson 4B Lesson 4J Lesson 4F Lesson 4G Lesson 4C Lesson 4 -- Review of Arithmetic~Numerals and Numbers • Natural or Counting Numbers ~Real Numbers • Multiplication & Division~Symbols of Equality and Inequality • Basic Operations ~Review of Operations With Decimal Numbers~Unit Multipliers • Conversions of Length

  2. Lesson 4.A Numerals and Numbers A number is an idea. A numeral is a single symbol or collection of symbols that we use to express the idea of a particular number. If we want to write something on paper to represent a number, we must use a numeral. Suppose we want to use a symbol to designate the idea of three. We could use any of the following: A number is an idea! Thus, the value of a numeral is the number represent by the numeral, and we see that the words value and number have the same meaning.

  3. Lesson 4.B Natural or Counting Numbers The system of numeration we use today to designate numbers is called the decimal system. It was originally invented by the Hindus in India. It was passed to their Arab neighbors where it picked up its name as the Arabic Numeration System and finally found its way to Europe around A.D. 1200. The decimal system uses 10 symbols that we call digits. We call the numbers that we use to count objects or things the natural numbers or counting numbers. When we begin counting, we always begin with the number 1 and follow it with the number 2, etc. We represent the set of natural numbers using the set notation below: The three dots after the 3 indicates that the list continues without end.

  4. Lesson 4.C Real Numbers The numbers of arithmetic are zero and the positive real numbers. We say that a positive real number is any number that can be used to describe a physical distance greater than zero. Below are some examples of positive real numbers: The number zero is not a positive number, but it can be used to describe a physical distance of no magnitude. Therefore, we also call it a real number. In addition to the positive numbers and zero, in algebra we use numbers that we call negative numbers, and these numbers are also called real numbers.

  5. Lesson 4.CPage 2 Real Numbers(Continued) The ancients did not understand the concept of negative numbers or zero. However, for modern mathematicians, physicists, or chemists, the idea or concept of negative numbers is a useful concept. We say that every positive real number has a negative counterpart, and we call these numbers the negative real numbers. In order to designate a negative number, we must always use a minus or negative sign, as we see below by writing a negative seven. We can represent a positive 7 either with the plus sign or no sign. We must remember that when we write a numeral with no sign, we designate a positive number.

  6. Lesson 4.D Number Lines To construct a number line, we first draw a line and divide it into equal units of lengths. 0 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 We choose a point on the number line as our base point. We call this beginning point the origin. Then we associate the positive real numbers with the points to the right of the origin and the negative real numbers with the points to the left of the origin.

  7. Lesson 4.DPage 2 Number Lines(Continued) 0 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 On the number line above we have indicated the location of zero, the counting numbers, and the negative counterpart of each counting number. We can now indicate the location of any real number by locating it in relation to the numbers shown. Suppose we want to locate -5½, + 2¾, and +8.6 on the number line above. When we place a dot on the number line to indicate the location of a number, we say we have graphed the number and that the dot is the graph of the number.

  8. Lesson 4.E Multiplication and Division of Fractions

  9. Lesson 4.EPage 2 Multiplication and Division of Fractions (Continued)

  10. Lesson 4.F Symbols of Inequality We use the equal sign (=) to designate two quantities that are equal. We use the symbol ≠ to designate two quantities that are not equal.

  11. Lesson 4.G Basic Operations The basic operations of arithmetic and algebra are addition, subtraction, multiplication, and division. Addition When we add two or more numbers together we call each number an addend. The answer to an addition problem is called the sum. We remember that when we add zero to any particular real number the sum is the real number itself. a + 0 = 0Additive Property of Zero

  12. Lesson 4.GPage 2 Basic Operations (Continued) Subtraction In subtraction of two numbers, we call the first number the minuend; the second number the subtrahend; and the result, the difference. minuend difference subtrahend

  13. Lesson 4.GPage 3 Basic Operations (Continued) Multiplication In multiplication of numbers, the numbers that we multiply are called factors and the result is called the product. When we multiply any particular number by 1 the product is the particular number itself. The product of any real number and the number zero is the number zero.

  14. Lesson 4.GPage 4 Basic Operations (Continued) Division In division of numbers, the first number or the number on top is called the dividend and the second number or the number on the bottom is called the divisor. dividend divisor The answer to a division problem is called the quotient.

  15. 4.06 x 0.016 ) 0.03 6.039 Lesson 4.H Decimal Numbers Example 4.1 Add 4.0016 and 0.02163. 3 2 4 0 2 3 Example 4.2 Subtract 0.02163 from 4.0016. 3 9 7 9 9 7 Example 4.3 Multiply 4.06 X 0.016. Example 4.3 Divide 6.039 by 0.03.

  16. Lesson 4.I Unit Multipliers Examples of unit multipliers.

  17. Lesson 4.J Conversion of Length Example 4.5 Use one unit multiplier to convert 32 feet to inches Hint (1 ft = 12 in) Example 4.6 Use one unit multiplier to convert 36 feet to mi. Hint (5280 ft = 1 mi.)

  18. Lesson 4.J Conversion of Length (Continued) Example 4.7 Use one unit multiplier to convert 47.25 inches to centimeters. Hint (1 in = 2.54 cm) Example 4.8 Use one unit multiplier to convert 42 meters to cm. Hint (1 m = 100 cm)

  19. Lesson 4.J Conversion of Length (Continued) Example 4.9 Use two unit multipliers to convert 42 feet to centimeters. Hint:(We will first convert feet to inches and then convert inches to centimeters.) Example 4.10 Use two unit multipliers to convert 4 miles to inches. Hint:(We will first convert miles to feet and then convert feet to inches.)

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