module 2: demonstrate an understanding of the use …

NC Business Administration Services NQF 3: SAQA ID 67465 US 9010

of 14 Business Administration Services NQF 3 – BA3 – LG 9 – US 9010 Issue 3: 01-01-2020

MODULE 2:

DEMONSTRATE AN UNDERSTANDING OF THE USE OF DIFFERENT NUMBER

BASES AND MEASUREMENT UNITS AND AN AWARENESS OF ERROR IN THE

CONTEXT OF RELEVANT CALCULATIONS

SAQA 9010

People credited with this unit standard are able to:

Convert numbers between the decimal number system and binary number system

Work with numbers in different ways to express size and magnitude.

Demonstrate the effect of error in calculations.

2.1 CONVERT NUMBERS BETWEEN THE DECIMAL NUMBER SYSTEM AND THE

BINARY NUMBER SYSTEM

The decimal (base ten or occasionally denary) numeral system has ten as its base. It is the

most widely used numeral system, perhaps because humans have four fingers and a thumb

on each hand, giving a total of ten digits over both hands

Decimal notation is the writing of numbers in the base-ten numeral system, which uses

various symbols (called digits) for no more than ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8

and 9) to represent any numbers, no matter how large. These digits are often used with a

decimal separator which indicates the start of a fractional part, and with one of the sign

symbols + (positive) or − (negative) in front of the numerals to indicate sign.

Binary-coded decimal (BCD) is an encoding for decimal numbers in which each digit is

represented by its own binary sequence. Its main virtue is that it allows easy conversion to

decimal digits for printing or display and faster decimal calculations. Its drawbacks are the

increased complexity of circuits needed to implement mathematical operations and a

relatively inefficient encoding – 6 wasted patterns per digit. Even though the importance of

BCD has diminished, it is still widely used in financial, commercial, and industrial

applications. In BCD, a digit is usually represented by four bits which, in general, represent

the values/digits/characters 0-9. Other bit combinations are sometimes used for sign or other

indications. To BCD-encode a decimal number using the common encoding, each decimal

digit is stored in a four-bit nibble.

Decimal: 0 1 2 3 4 5 6 7 8 9

BCD: 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001

Thus, the BCD encoding for the number 127 would be: 0001 0010 0111

Since most computers store data in eight-bit bytes, there are two common ways of storing

four-bit BCD digits in those bytes:

each digit is stored in one byte, and the other four bits are then set to all zeros, all ones

(as in the EBCDIC code), or to 0011 (as in the ASCII code)

two digits are stored in each byte.



Unlike binary encoded numbers, BCD encoded numbers can easily be displayed by

mapping each of the nibbles to a different character. Converting a binary encoded number to

decimal for display is much harder involving integer multiplication or divide operations. The

BIOS in many PCs keeps the date and time in BCD format, probably for historical reasons (it

avoided the need for binary to ASCII conversion).

Example: 5632 = 5 x 1000 + 6 x 100 + 3 x 10 + 2 x 1

= 5 x 103 + 6 x 102 + 5 x 101 + 2 x 100

In the decimal system there are 10 different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Given a decimal number, how do we find the binary equivalent?

Binary is a base two system, so we need to know which powers of 2 are contained in the

decimal number. Let’s take the number 241:

Step 1: Divide by two. Since the number 241 is indivisible, we write a 1. Then we subtract 1

from 241 to get 240 and divide by two to get 120.

1

Step 2: Divide by two. Since 120 is divisible, we write a 0 to the left of the previous number

and divide by two to get 60.

01



001



0001

Step 5: Divide by two. Since 15 is indivisible, we write a 1 to the left of the previous number.

Then we subtract 1 from 15 to get 14 and when we divide we get 7.

10001

Step 6: Divide by two. Since 7 is indivisible, we write a 1 to the left of the previous numbers.


110001



1110001


Then we subtract 1 from 1 to get 0, which means we are finished.

11110001

To Summarize the Steps:

Step 1: See if the number is divisible by 2. If it is write 0 and divide by 2 to get the next

number. If the number, on the other hand, is indivisible by 2, write a 1, then subtract 1 from

the number and divide by two to get the next number.

Step 2: Continue this process, writing each 1 or 0 to the left of the previous 1 or 0, until the

number which we are left to divide by is 0.

Step 3: You should always check your answer to make sure that your conversion is correct.



Class Activity: In groups Please follow the instructions from the facilitator to complete the formative activity in your Learner Workbook

In your groups, try and convert the following from decimal to binary: 313

STEPS EXPLANATION OF STEPS (EXPLAIN HOW YOU HAVE DONE IT,

ACCORDING TO THE STEPS GIVEN)

YOUR

ANSWER

Step

1

Step

2

Step

3

Step

4

Step

5

Step

6

Step

7

Step

8



Given a binary number, how do we find the decimal equivalent?

Binary numbers use base 2. There are only 2 digits: 0 and 1.

Example: (1011)2 = 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20

= 1 x 8 + 0 x 4 +1 x 2 + 1 x 1

= (11)10

Therefore (1011)2 is the binary representation for eleven

Let's first look at a number system we are all familiar with, the Decimal Number system:

In Decimal the number:

We can re-write this using scientific notation

We can use a similar process when converting binary numbers into decimal numbers:

Therefore, we know that the binary number 1010 is equivalent to the decimal number 10.

You should notice that while the decimal number system has a ones place, tens place,

hundreds place, thousands place, and so on, the binary number system has a ones place,

twos place, fours place, eights place, and so on, respectively.

Another Binary to Decimal Example:


In your groups, try and convert the following from binary to decimal: 0100 0100



2.2 WORK WITH NUMBERS IN DEFFERENT WAYS TO EXPRESS SIZE AND MAGNITUDE

SI Units Since measurements cross international borders, so must measurement standards. The International Bureau of Weights and Measures determines, conducts and monitors international metrological research. The units of measurement are the accepted International System of Units, or SI, which is based on the metric system. Scientists throughout the world use the SI units for their measurements. Only the United States of America still uses traditional units in addition to SI units. Just like there are rules for grammar and spelling, there are specific rules for writing SI units. For example, when writing a quantity and a unit, there is a space between the two, such as 15 m, not 15m.

Base and derived Units There are seven base SI units.

Physical Quantity Name of Unit Abbreviation

Mass kilogram kg

Length meter m

Temperature Kelvin K

Amount of substance

Mole mol

Time second s

Electric current ampere A

Luminous intensity candela cd

There are several SI units that can be expressed in terms of one or more of the seven base SI units listed in the above table, these are derived SI units.

Quantity Definition Derived unit name

Area length times length m2

Volume area times length m3

Force mass times acceleration

(kg x m)/s2 (Newton, N) Newton: a unit of force, defined as that force which gives a mass of one kilogram an acceleration of one meter per second per second

Energy force times distance

(kg x m2)/s2 (Joule, J) The joule, a unit of energy, defined as the work done when the point of application of a Newton is displaced one meter in the direction of the force

A problem with the SI units of measurement is that very large numbers or very small numbers must often be used to express many measurements. When the size of the numbers become cumbersome, then scientific notation is used.



Scientific Notation Scientific notation makes perceiving very large and very small numbers much easier. For example, there are about 602 200 000 000 000 000 000 000 atoms in one mole of a substance. The same number can be expressed in scientific notation as 6.022 x 1023.

The definition of a mole. If 10-3 moles are a millimole, if 10-6 moles is a micromole, then what is a "guacamole"?

Not only is this easier to write, but it gives us an immediate idea of just how large this number is. The exponent, the small-size number in the upper right-hand corner of the 10, is 23. The exponent of 23 tells us immediately that there are 23 place-holding digits to the right of the first digit of the coefficient. In this example, 6.022 is the coefficient. To write very small numbers, such as the diameter of a carbon atom, we use negative exponents. The diameter of a carbon atom is about 0.0 000 000 001 m. In scientific notation, a carbon atom is 1.0 x 10-10 m in diameter. When converting numbers to scientific notation follow these simple rules: 1. Determine the power of ten:

o For numbers greater than or equal to 1, add the number of places to the right of the first digit. (Note: it is possible to have a 0 exponent,100=1).

For example, there are 7 places to the right of the first digit in the number, 12 000 000 and the power of ten is 107.

o For numbers less than 1, subtract the number of places to the left of the first

nonzero digit.

For example, there are 7 places to left of the first non-zero digit in the number, 0.00 000 012 and the power of ten is 10-7.

No matter which SI unit is used to report a measurement, there is an element of uncertainty. Every measurement is limited by the reliability of the measuring instrument and the skill of the operator.



Metric Conversions The metric unit selected for use is based upon the size of the object that is being measured. Often however, the quantity used must be converted to a different size unit. SI unit conversion can be done by simply moving the decimal to increase or decrease the size of the number. Then the appropriate prefix for the new unit must be given. Figure 3: Metric unit conversion is simply a matter of moving the decimal in the correct direction and the correct number of places. Unit conversion is done by taking the numerical difference between the exponents and moving the decimal in the appropriate direction. If the new unit is smaller, then you will need more of them, so the decimal will be moved to the right. If the new unit is larger, you will need fewer of them, so the decimal will be moved to the left. Precision and Accuracy Precision and accuracy are often used as synonyms but when referring to uncertainty in measurement each term means something different. It is possible for a measurement to be precise or accurate or both or neither. If the measurement is close to repeated measurements of the same value, then it is considered precise. If the measurement is close to the true value (or an expected value if the true value is unknown) then it is considered accurate. Precise numbers are often also accurate, but it is possible to repeatedly measure the same error and come up with precise, but inaccurate numbers. Precision is how closely

repeated measurements match each other.

Accuracy is how closely a measurement matches the correct or expected value.

A ball player's height can be measured precisely, but still not be accurate.

What is size? If you look up "size" in the dictionary, it will say something like: "The physical magnitude, extent, or bulk of something." The basic dimensions of measurement are length, mass and time. The derived dimensions are area, force, etc. For our purposes, we will focus primarily on length, though we will also discuss the others briefly. It is fairly simple to imagine the length of a meter, just look at the length of your arm. You can walk 1000 meters in about 15 minutes, or drive the same distance in about a minute. However, when lengths get longer than what we can personally experience, such as the distance from the Earth to the sun, or the diameter of the universe, it becomes difficult to comprehend. Our eyes generally give us the ability to discern things as small in size as an ant's eye and the point of a pin. About 300 years ago, with the invention of the light microscope it was discovered that there is a whole new world of living organisms much smaller than a pinpoint.



What is scale? One way to think of scale is the relationship between the actual length you are measuring, and the way that length is represented numerically or visually. A scale has a succession of ascending and descending steps, or relative dimensions, used to assess the absolute or relative size of some property of an object, such as length, temperature, or mass. The unit of measurement for measuring length can be inches, feet, rods, meters or one of many other traditional units of length. The unit of measurement for temperature can be degrees Fahrenheit, degrees Celsius or Kelvin. Weight can be measured in pounds, grams, tons or other units of mass. Scales can range from smaller than an atom to larger than the universe and hence, a linear scale is not a convenient representation. A logarithmic scale, however, uses the Power of 10 to represent and compare the relative size or distances of objects with actual lengths that are so drastically different that it would be difficult to represent them on a linear scale.

In a linear scale, the lengths represented between each of the two equi-distant marks are equal. So, the distance between 1 and 2 is the same as the distance between 3 and 4. The map scale is linear scale. Remember that each step, in a logarithmic scale, differs by one order of magnitude from its preceding or the succeeding step. Look at the illustration of a logarithmic scale below.

In this scale the length represented between 1 and 2 is 10 times longer than the length between 0 and 1. Similarly, the length represented between 2 and 3 is 10 times longer than the length represented between 1 and 2 and 100 times longer than the length represented between 0 and 1. Each step in the logarithmic scale is an order of magnitude. Little by little How small is small? It depends on how small you are. A bee is small. You can squish one with the tip of your finger, if you are careful! But compared to a pollen grain collected by the bee, then the bug is big! In fact, a bee is considered to be at the macro-scale, the pollen grain, which can be seen only with the aid of a microscope, is at the micro-level. This is getting down there. So how low do we need to go until we are talking small? How about the size of the pore on the pollen grain? Are we talking small yet? Yeah, but we are still not at the nano-scale! The nano-scale begins at 10 nanometres in length and a nanometre is one-billionth of a meter.

"What the scale?" Things at different scales can look very similar. So you may not be able to tell what it is, if you don't know the scale. Also, some things look very different at different scales, which can also be confusing. It is helpful to know what the scale is for something you are studying.



(A) A coin. (B) The word "TRUST" on the coin. (C) The letter "R" in the word "TRUST". If you saw only the microscopic image of the

letter "R", would you know what you were looking at? However, if you were given the scale, 0.7 mm, you would at least know you were looking at something about the size of the letter "R".

Temperature There have been five main temperature scales, each one being named after the person who invented it. G D FAHRENHEIT (1686-1736) a German physicist, in about 1714 proposed the first practical scale. He called the freezing-point of water 32 degrees (so as to avoid negative temperatures) and the boiling-point 212 degrees. Anders CELSIUS (1701-1744) a Swedish astronomer, proposed the 100-degree scale (from 0 to 100) in 1742. This was widely adopted as the centigrade scale. But since grades and centigrade were also measures of angle, in 1947 it officially became the Celsius scale. Also, the S I system of units gives preference to naming units after people where possible. Nowadays, while scientists use the KELVIN scale, the CELSIUS scale is the preferred scale in our everyday lives. However, the Fahrenheit scale is still widely used and there frequently is a need to be able to change from one to the other.

To change temperature given in Fahrenheit (F) to Celsius (C) Start with (F); subtract 32; multiply by 5; divide by 9; the answer is (C) To change temperature given in Celsius (C) to Fahrenheit (F) Start with (C); multiply by 9; divide by 5; add on 32; the answer is (F)


In your groups, convert the following temperatures, using the formulae given above: (Remember to show your calculations!)

UNITS TO BE

CONVERTED YOUR CALCULATION YOUR ANSWER

39°C to F

18°C to F

4°C to F

106°F to °C

31°F to °C

155°F to °C



Length The S I unit of length is the metre. To change any of these other units of length into their equivalent values in metres use the operation and conversion factor given. So 1 metre is equivalent to:

centimetres x 0.01 feet x 0.3048 inches x 0.0254 kilometres x 1000 leagues x (4000 to 5000) light years x 9 460 500 000 000 000 metres [m] 1 miles (UK and US) x 1609.344 miles (nautical) x 1852 yards x 0.9144


In your groups, convert the following lengths, using the formulae given above: (Remember to show your calculations!)

UNITS TO BE

CONVERTED YOUR CALCULATION YOUR ANSWER

15 meters to feet

22 meters to kilometres

18 meters to yards

3028 meters to miles (N)

2001 meters to miles

2.3 DEMONSTRATE THE EFFECT OF ERROR IN CALCULATIONS So what is the difference between rational and irrational numbers? Before we talk about rational and irrational numbers, let's make clear one other definition. An INTEGER is in the set: {...-3, -2, -1, 0, 1, 2, 3, ...} It is just a positive or negative whole number. Thus 454564 is an integer, but 1/2 isn't. Now, a rational number is any number that can be written as a ratio of two integers (hence the name!). In other words, a number is rational if we can write it as a fraction where the numerator and denominator are both integers. Now then, every integer is a rational number, since each integer n can be written in the form n/1. For example, 5 = 5/1 - thus 5 is a rational number. However, numbers like 1/2, 45454737/2424242, and -3/7 are also rational since they are fractions where the numerator and denominator are integers. An irrational number is any real number that is not rational. By "real" number I mean, loosely, a number that we can conceive of in this world, one with no square roots of negative numbers (numbers where square roots of negative numbers are involved are called complex, and there is lots of neat stuff there, if you are curious).



A real number is a number that is somewhere on your number line. So, any number on the number line that isn't a rational number is irrational. For example, the square root of 2 is an irrational number because it can't be written as a ratio of two integers. How would you imagine we would show something like that? The proof is a proof by contradiction. We assume that the square root of 2 CAN be written as p/q for some integers, p and q, and we get a contradiction. Other irrational numbers include: square root of 3, the square root of 5, pi, e... There are lots of neat properties of rational numbers, irrational numbers and real numbers. For instance, it turns out that if you were to try to gauge how many rational numbers, irrational numbers, and real numbers there are between 0 and 1, you would find that while there are infinitely many of each kind of number, there are many, many more irrational numbers than rational numbers. The sizes of the infinities involved are somehow a little different. Another property is that between any two rational numbers on the number line there is an irrational number; also, between any two irrational numbers there is a rational number.

What is an integer? {... -3, -2, -1, 0, 1, 2, 3, ...} Integers are the whole numbers, negative whole numbers, and zero. For example, 43434235, 28, 2, 0, -28, and -3030 are integers, but numbers like 1/2, 4.00032, 2.5, Pi, and -9.90 are not. We can say that an integer is in the set: {...3, -2, -1, 0, 1, 2, 3,} (the three dots mean you keep going in both directions.) It is often useful to think of the integers as points along a 'number line', like this:

Note that zero is neither positive nor negative.

About integers The terms even and odd only apply to integers; 2.5 is neither even nor odd. Zero, on the other hand, is even since it is 2 times some integer: it's 2 times 0. To check whether a number is odd, see whether it's one more than some even number: 7 is odd since it's one more than 6, which is even. Another way to say this is that zero is even since it can be written in the form 2*n, where n is an integer. Odd numbers can be written in the form 2*n + 1. Again, this lets us talk about whether negative numbers are even and odd: -9 is odd since it's one more than -10, which is even. Every positive integer can be factored into the product of prime numbers, and there's only one way to do it for every number. For instance, 280 = 2x2x2x5x7, and there's only one way to factor 280 into prime numbers. This is an important theorem: The Fundamental Theorem of Arithmetic. When most mathematicians are talking to each other, they use Z to refer to the set of integers. In German the word "zahlen" means "to count" and "Zahl" means "number." Mathematicians also use the letter N to talk about the set of positive integers, in other words the set {1,2,3,4,5,6, ...}. Rational Numbers 5/1, 1/2, 1.75, -97/3 ... A rational number is any number that can be written as a ratio of two integers (hence the name!). In other words, a number is rational if we can write it as a fraction where the numerator and denominator are both integers.



The term "rational" comes from the word "ratio," because the rational numbers are the ones that can be written in the ratio form p/q where p and q are integers. Irrational, then, just means all the numbers that aren't rational. Every integer is a rational number, since each integer n can be written in the form n/1. For example, 5 = 5/1 and thus 5 is a rational number. However, numbers like 1/2, 45454737/2424242, and -3/7 are also rational, since they are fractions whose numerator and denominator are integers. So the set of all rational numbers will contain the numbers 4/5, -8, 1.75 (which is 7/4), -97/3, and so on. Is .999 repeating a rational number? Well, a number is rational if it can be written as A/B (A over B): .3 = 3/10 and .55555.... = 5/9, so these are both rational numbers. Now look at .99999999.... which is equal to 9/9 = 1. We have just written down 1 and .9999999 in the form A/B where A and B are both 9, so 1 and .9999999 are both rational numbers. In fact, all repeating decimals like .575757575757..., all integers like 46, and all finite decimals like .472 are rational. Irrational Numbers: sqrt (2), pi, e, the Golden Ratio ...

Irrational numbers are numbers that can be written as decimals but not as fractions. An

irrational number is any real number that is not rational. By real number we mean, loosely, a

number that we can conceive of in this world, one with no square roots of negative numbers

(such a number is called complex.) A real number is a number that is somewhere on a

number line, so any number on a number line that isn't a rational number is irrational. The

square root of 2 is an irrational number because it can't be written as a ratio of two integers.

Other irrational numbers include the square root of 3, the square root of 5, pi, e, and the

golden ratio. Pi is an irrational number because it cannot be expressed as a ratio (fraction) of

two integers: it has no exact decimal equivalent, although 3.1415926 is good enough for

many applications.

The square root of 2 is another irrational number that cannot be written as a fraction. In

mathematics, a name can be used with a very precise meaning that may have little to do

with the meaning of the English word. ("Irrational" numbers are NOT numbers that can't

argue logically!)

Scientific Notation in numbers

Chemists often work with numbers that are extremely large or extremely small. For example,

there are 10,300,000,000,000,000,000,000 carbon atoms in a 1-carat diamond each of

which has a mass of 0.000,000,000,000,000,000,000,020 grams. It is impossible to multiply

these numbers with most calculators because they can't accept either number as it is written

here. To do a calculation like this, it is necessary to express these numbers in scientific

notation, as a number between 1 and 10 multiplied by 10 raised to some exponent.

Exponent Review

Some of the basics of exponential mathematics are given below.

Any number raised to the zero power is equal to 1. 1�= 1 10�= 1

Any number raised to the first power is equal to itself. 11 = 1 101 = 10



Any number raised to the nth power is equal to the product of that number times itself n-1

times.

22 = 2 x 2 = 4 105 = 10 x 10 x 10 x 10 x 10 = 100,000

Dividing by a number raised to an exponent is the same as multiplying by that number raised

to an exponent of the opposite sign.

Converting to Scientific Notation

The following rule can be used to convert numbers into scientific notation: The exponent in

scientific notation is equal to the number of times the decimal point must be moved to

produce a number between 1 and 10. In 1990 the population of Johannesburg / Pretoria was

6,070,000 �1000. To convert this number to scientific notation we move the decimal point to

the left six times.

6,070,000 = 6.070 x 106

To translate 10,300,000,000,000,000,000,000 carbon atoms into scientific notation, we move

the decimal point to the left 22 times.

10,300,000,000,000,000,000,000 = 1.03 x 1022

To convert numbers smaller than 1 into scientific notation, we have to move the decimal

point to the right. The decimal point in 0.000985, for example, must be moved to the right

four times.

0.000985 = 9.85 x 10-4

Converting 0.000,000,000,000,000,000,000,020 grams per carbon atom into scientific

notation involves moving the decimal point to the right 23 times.

0.000,000,000,000,000,000,000,020 = 2.0 x 10-23

The primary reason for converting numbers into scientific notation is to make calculations

with unusually large or small numbers less cumbersome. Because zeros are no longer used

to set the decimal point, all of the digits in a number in scientific notation are significant, as

shown by the following examples.

2.4 x 1022 2 significant figures

9.80 x 10-4 3 significant figures

1.055 x 10-22 4 significant figures


In your groups, link the number on the left to their scientific notation numbers in the right

column by drawing a line to the matching number



0.004694

1.98 x 100

19.8 x 100

1.98 4.679 x 106

4.679 x 105

4,679,000 4.694 x 10-3

46.94 x 10-4

Do you know this number, 300,000,000 m/sec.?

It's the Speed of light!

Do you recognize this number, 0.000 000 000 753 kg.?

This is the mass of a dust particle!

Scientists have developed a shorter method to express very large numbers. This method is

called scientific notation. Scientific Notation is based on powers of the base number 10.

The number 123,000,000,000 in scientific notation is written as:

The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less

than 10.

The second number is called the base. It must always be 10 in scientific notation. The base

number 10 is always written in exponent form. In the number 1.23 x 1011 the number 11 is

referred to as the exponent or power of ten.

To write a number in scientific notation:

Put the decimal after the first digit and drop the zeroes.

In the number 123,000,000,000. The coefficient will be 1.23. To find the exponent count the

number of places from the decimal to the end of the number. In 123,000,000,000 there are

11 places. Therefore, we write 123,000,000,000 as:

Exponents are often expressed using other notations. The number 123,000,000,000 can

also be written as:

1.23E+11 or as 1.23 X 10^11

For small numbers we use a similar approach. Numbers smaller than 1 will have a negative

exponent. A millionth of a second is: 0.000001 sec. or 1.0E-6 or 1.0^-6 or

module 2: demonstrate an understanding of the use …

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