Chapter 1
Basic Mathematics

Arithmetic

Fractions

A fraction is a ratio of two numbers . The fraction bar means the operation of division.

=

Mixed numbers and improper fractions:

This is a mixed number - .

This is an improper fraction - .

In the long run improper fractions are easier to work with then mixed numbers. The way to change a mixed number into an improper fraction is to take the integer in front of the fraction and multiply it by the denominator then add it to the numerator. This number is now your new numerator. The denominator remains the same.

Example 1:

For the mixed number above, , you would take then add and finally the new fraction would be .

Now we look at the four operations on fractions: addition, subtraction, multiplication, and division.

The biggest thing to remember about adding and subtracting fractions is that the fraction must have the same denominator before you can add or subtract them.

Example 2:

We will add the following fractions together:

First we need to decide what the common denominator should be to add the fractions. One way to find the common denominator is to multiply the two denominators together. Which for this example is . So we need to have both denominators be 12. We need to multiply the first fraction, numerator and denominator, by 3. The second fraction needs to be multiplied by to get a denominator of 12.

Then the new fractions are:

Now we add the numerators together and put that answer over the denominator.

Example 3:

Subtract the following fractions:

The common denominator is 21.

Now we subtract the numerators:

Example 4:

A common denominator is 18, but the least common denominator is 6.

Here is a link to an enhanced podcast on adding and subtracting fractions.

Podcast on Adding and Subtracting Fractions

Multiplication and Division of Fractions:

When we multiply fraction we multiply the numerators together and multiply the denominators together, we then reduce to lowest terms.

Example 5:

Multiply the following fractions:

This fraction cannot be reduced.

Example 6:

Multiply the following fractions:

This fraction can be reduced before the multiplication:

Then multiply just like the previous example:

This process is called cross cancelling before multiplying.

When dividing fractions the most common algorithm is to take the reciprocal of the second fraction and then multiply the fractions as above. The reason this works is illustrated below:

Example 7:

Divide the following fractions:

First we take the reciprocal of the second fraction, then multiply:

Here is a link to an enhanced podcast on mulitipling and dividing fractions.

Podcast on Multiplying and Dividing Fractions

Decimals

Converting Fractions to Decimals

When converting from a fraction to a decimal, the fraction bar means division. You may either divide by long division or with a calculator.

Example 8:

Convert into a decimal.

If the denominator is a power of 10, then the decimal equivalent can be found by moving the decimal point the number of zeros in the denominator.

Example 9:

Podcast on Decimals

Significant Figures

Rules for determining whether zeros are significant figures

1. Zeros between other nonzero digits are significant.

2. Zeros in front of nonzero digits are not significant.

3. Zeros that are at the end of a number and also to the right of the decimal point are significant.

4. Zeros at the end of a number but to the left of a decimal are significant if they have been measured or are the first estimated digit; otherwise, they are not significant.

Rules for calculating with significant figures

 Type of Calculation Rule Addition or Subtraction The final answer should have the same number of digits to the right of the decimal as the measurement with the smallest number of digits to the right of the decimal. Multiplication or Division The final answer has the same number of significant figures as the measurement having the smallest number of significant figures.

Example 10:

Using the rules above we need to round to the tenth place since that is the least precise measurement. So the final answer would be 250.7.

Example 11:

Using the rules above we see that the first number has 5 significant figures and the second number has 3. Therefore, we need 3 significant figures in our answer. So the final answer is 15.5.

Podcast on Significant Digits

Scientific Notation

When very large or very small numbers are used in Physics it is convenient to use scientific notation.

Example 12:

To write the number 9192631700 in scientific notation it you count the number of decimal places so that the decimal is to the right of the last number on the left. So the answer would be .

Example 13:

If the number is very small for example 0.0000000000000000001602 C then the exponent is negative. This number is .

Podcast on Scientific Notation

Dimensional Analysis

Convert the following:

Example 14:

15 inches to yards

Example 15:

60 miles/hour to feet/second

Podcast on Dimensional Analysis

Algebra

Absolute Value – The absolute value can be regarded as the distance of a number from zero.

Integers

Example 16:

Example 17:

Podcast on Integers

Order of Operations

1. Perform all operations inside grouping symbols
2. Exponents
3. Multiplication and Division from the left
4. Addition and Subtraction from the left

Example 18:

Podcast on Order of Operations

Algebraic Expressions

Example 19:

Exponents

When you use exponents in problems you have to follow rules when multiplying and dividing numbers that use exponents.

When you multiply numbers with exponents you add the exponents:

When you divide numbers with exponents you subtract the exponents:

When you have a power raised to a power you multiply the exponents:

Example 20:

Podcast on Exponents

Evaluating Algebraic Expressions

Example 21:

Evaluate the following expression if a = 5, b = 3, and c = -2.

Equations

Example 22:

Solve the following equation for x.

Variation

Direct Variation –When two variable quantities have a constant (unchanged) ratio, their relationship is called a direct variation. It is said that one variable "varies directly" as the other. The constant ratio is called the constant of variation.

The general equation for direct variation is y = kx.

Inverse Variation – A relationship between two variables in which the product is a constant. When one variable increases the other decreases in proportion so that the product is unchanged.

The general equation for inverse variation is .

Podcast on Algebra

Units of Measurement

SI Units of Measurement

 Base Units Quantity Unit Name Symbol Mass kilogram kg Length meter m Time second s Electric Current ampere A Temperature kelvin K Amount of Substance mole mol Luminous Intensity candela cd

 Derived Units Quantity Unit Name Symbol British Units Absorbed Dose gray Gy rad Charge coulomb C esu Electric Potential volt v Dose Equivalent sievert Sv rem Energy joule J erg Exposure coulomb/kilogram C/kg roentgen Frequency hertz Hz cycles/second Force newton N Magnetic Flux weber Wb Magnetic Flux Density tesla T gauss Power watt W Radioactivity bequerel Bq curie

Metric Prefixes

 Factor Prefix Symbol yotta- Y zetta- Z exa- E peta- P tera- T giga- G mega- M kilo- k hecto- h deka- da deci- d centi- c milli- m micro- µ nano- n pico- p femto- f atto- a zepto- z yocto- y

The Prefixes that you need to know are the ones with the yellow background.

Commonly Used SI Prefixes

Unit Symbol Meaning
kilovolt
kV
centimeter
cm
milliampere
mA
milligray
mGy
nanosecond
ns

Units in Radiology – Customary Units

Roentgen: A basic unit of measurement of the ionization produced in air by gamma or x-rays. One Roentgen (R) is exposure to gamma or x-rays that will produce one electrostatic unit of charge in one cubic centimeter of dry air.
It is a measure of the ionizations of the molecules in a mass of air. The main advantage of this unit is that it is easy to measure directly, but it is limited because it is only for deposition in air, and only for gamma and x rays.

rad: A unit of measurement of any kind of radiation absorbed by humans. This relates to the amount of energy actually absorbed in some material, and is used for any type of radiation and any material. The unit rad can be used for any type of radiation, but it does not describe the biological effects of the different radiations. One rad is equal to the absorption of 100 ergs of radiation energy per gram of material.

rem: The roentgen equivalent man or rem is the unit of effective radiation dose. The rem is a unit used to derive a quantity called equivalent dose. This relates the absorbed dose in human tissue to the effective biological damage of the radiation. Not all radiation has the same biological effect, even for the same amount of absorbed dose. Equivalent dose is often expressed in terms of thousandths of a rem, or mrem. To determine equivalent dose (rem), you multiply absorbed dose (rad) by a quality factor (Q) that is unique to the type of incident radiation.

curie: The original unit used to describe the intensity of radioactivity in a sample of material. One curie equals thirty-seven billion (37,000,000,000) disintegrations per second, or approximately the radioactivity of one gram of radium.

Units of Radiology – SI Units

Air Kerma: air kerma means kerma in a given mass of air. The unit used to measure the quantity of air kerma is the Gray (Gy). For X-rays with energies less than 300 kiloelectronvolts (keV), 1 Gy = 100 rad. In air, 1 Gy of absorbed dose is delivered by 114 roentgens (R) of exposure. Kerma means the sum of the initial energies of all the charged particles liberated by uncharged ionizing particles in a material of given mass.

Gray: radiation energy deposited in material divided by the mass of the material. An often-used unit for absorbed dose is the gray (Gy). One gray is equal to one joule of energy deposited in one kg of a material. The unit gray can be used for any type of radiation, but it does not't describe the biological effects of the different radiations. Absorbed dose is often expressed in terms of hundredths of a gray, or centi-grays. One gray is equivalent to 100 rads. (gray=Gy=J/kg)

Seivert: When absorbed dose is adjusted to account for the amount of biological damage a particular type of radiation causes, it is known as dose equivalent. This relates the absorbed dose in human tissue to the effective biological damage of the radiation. Not all radiation has the same biological effect, even for the same amount of absorbed dose. Equivalent dose is often expressed in terms of millionths of a sievert, or micro-sievert. To determine equivalent dose (Sv), you multiply absorbed dose (Gy) by a quality factor (Q) that is unique to the type of incident radiation. One sievert is equivalent to 100 rem. The SI unit for dose equivalent is called the seivert (Sv).

Becquerel: The unit of radioactive decay equal to 1 disintegration per second. Often radioactivity is expressed in larger units like: thousands (kBq), one millions (MBq) or even billions (GBq) of a becquerels. 37 billion () becquerels = 1 curie (Ci)

End of Chapter 1 Lecture Notes