# Grade 6 Math Circles. Binary and Beyond

Size: px
Start display at page:

## Transcription

1 Faculty of Mathematics Waterloo, Ontario N2L 3G1 The Decimal System Grade 6 Math Circles October 15/16, 2013 Binary and Beyond The cool reality is that we learn to count in only one of many possible number systems: the decimal system. The decimal system is our default number system. The word decimal doesn t just mean a number with digits on the right side of a dot. In fact, the word decimal is derived from the Latin word decem which means ten. Strictly speaking, the decimal system is a number system of base 10. Don t worry about all the fancy language right now. Just follow along with the example below to find out what s so special about the number 10 in the decimal system. Example 1 Notice that we can write the number 4175 in the following way: 4175 = (4)(1000) + (1)(100) + (7)(10) + (5)(1) = (4)(10 3 ) + (1)(10 2 ) + (7)(10 1 ) + (5)(10 0 ) We can also write in a similar way: = (1)(10 4 ) + (2)(10 3 ) + (0)(10 2 ) + (9)(10 1 ) + (9)(10 0 ) Do you see a pattern? Each number can be written as a sum composed of their digits times specific powers of 10, where the powers correspond to the place value of the digits in the number. For example, in 4175, the digit 7 is in the tens column and so corresponds to the power So we can basically say that in the decimal system, numbers are built from powers of 10. 1

2 Let s summarize what we have discovered: Place Value Thousands Hundreds Tens Ones Power And note that this pattern continues for higher place values. Also, we should make one other important point about the decimal system. That is, in the decimal system, digits may be integers from 0 to 9 only. We cannot have a digit with a value greater than or equal to 10 since it would require at least two place values. So we can now talk about the decimal number system as as the following: The decimal system is a number system with base 10. Every place value corresponds to a power of 10 and a digit may be any integer from 0 to 9. Exercise I (a) Write the following numbers in expanded form (as sums of powers of 10): i ii. 593 iii (b) Write the following numbers in standard form (usual digit representation): i. (7)(10 4 ) + (4)(10 2 ) + (3)(10 1 ) ii. (9)(10 5 ) + (2)(10 3 ) + (8)(10 1 ) + (1)(10 0 ) + (4)(10 0 ) iii. (6)(10 3 ) + (4)(10 2 ) + (10)(10 1 ) + (3)(10 0 ) 2

3 Binary Numbers Another number system that is used every day is the binary system. It is the number system that computers use. It can be described as follows: The binary system is a number system with base 2. Every place value corresponds to a power of 2 and a digit may only be either 0 or 1. Some examples of binary numbers are 11, 10001, , and Note that these are not read as eleven, ten-thousand and one, etc., but as one-one, one-zero-zero-zero-one, and so on (to avoid confusion). Just like the decimal system used powers of 10 to create place values, the binary system uses powers of 2 to do the same. Look at the expanded form of the binary number and the way it is written in the place value chart: (1)(2 5 ) + (1)(2 4 ) + (0)(2 3 ) + (1)(2 2 ) + (1)(2 1 ) + (0)(2 0 ) Place Value Place Value Digit Converting From Binary to Decimal Example 2 Write the binary number as a decimal number. Solution First we represent the binary number with a place value chart: Place Value Place Value Digit Notice that we make 5 place value columns because the binary number is 5 digits long. We place the digits as they appear in the binary number. 3

4 Just as we did with the decimal numbers, we can now write as a sum of powers, in this case powers of (1)(2 4 ) + (0)(2 3 ) + (1)(2 2 ) + (0)(2 1 ) + (1)(2 0 ) = (1)(2 4 ) + (1)(2 2 ) + (1)(2 0 ) = = 21 So the binary number is the decimal number 21. Notice that all we actually need to do is add up the numbers in the Place Value row that have a 1 below in the Digits row. Lets summarize the steps to convert a binary number to a decimal number: 1. Write out the binary number using a place value chart (as in Example 2). 2. Add the numbers in the Place Value row that have a 1 below in the Digits row. 3. The sum is the decimal form of the binary number. Exercise II (a) Read the following binary numbers out-loud and then order them from least to greatest: i ii iii (b) Convert the following binary numbers to decimal numbers: i ii iii

5 Converting From Decimal to Binary Example 3 Write the decimal number 13 as a binary number. Solution We want to write 13 as a sum of powers of 2. Each power of 2 in the sum must be multiplied by a 1 or 0 since these are the only allowed integers for binary digits. What is the best way to do this? We can start by trying to find the greatest power of 2 that is less than or equal to 13. We find that this is 2 3 = 8, and so we may write 13 as the following: 13 = = (1)(2 3 ) + 5 Now we just need to write 5 as a sum of powers of 2. The greatest power of 2 that is less than or equal to 5 is 2 2 = 4. Therefore, we can write 13 = = (1)(2 3 ) + (1)(2 2 ) + 1 Now we only need to write 1 as a power of 2. Since 2 0 = 1, then 13 can be written as a sum of powers of 2 as 13 = (1)(2 3 ) + (1)(2 2 ) + (0)(2 1 ) + (1)(2 0 ) So now we can construct a binary place value chart with 8 and 2 3 in the left-most column in the Place Value rows: Place Value Place Value Digit Notice that we place the 1 s and 0 s in the Digits row so that they correspond to the appropriate powers of 2 (from the equation above). Therefore, we can just read off from the chart that 13 can be written in binary as

6 Now that you have seen an example of how to convert a decimal number to a binary number, you have one method to do it. But we can simplify the method in some easy to follow steps: 1. Find the greatest power of 2 less than or equal to the decimal number. 2. Make a place value chart with the left-most column containing the greatest power of 2 (from Step 1) in the Place Value rows. 3. Fill in the left-most column in the Digit row with a 1. Subtract the place value of this column from the original decimal number and keep track of the difference (called a running total). 4. Repeat Step 1 using the running total. Under that place value, put a 1 in the Digit row. Subtract the place value of this column from the running total to get a new running total. 5. Repeat the procedure until the running total becomes 0. Fill in the remaining spaces in the Digits column with zeros. 6. Read off the Digits row from left to right. This is the binary form of the decimal number. Example 4 Write the decimal number 26 as a binary number. Solution Step 1: The greatest power of 2 less than or equal to 26 is 2 4 = 16. Step 2: Construct the place value chart with the left-most column being the 2 4 place value: Place Value Place Value Digit Step 3: Running total = = 10. Place Value Place Value Digit 1 6

7 Step 4: The greatest power of 2 less than or equal to 10 is 2 3 = 8. Running total = 10 8 = 2. Place Value Place Value Digit 1 1 Step 5: The greatest power of 2 less than or equal to 2 is 2 1 = 2. Running total = 2 2 = 0. Place Value Place Value Digit Fill in the remaining spaces in the Digits row with zeros: Place Value Place Value Digit Step 6: The binary form of 26 is Notice that the running total is like a remainder in division. We are really just doing division. We are dividing the decimal number (and then the running total) by the greatest power of the base (in this case 2) that is less than or equal to the decimal number (or the running total in later steps). This idea will be important when we convert decimal numbers to numbers of an arbitrary base number system. Exercise III Write the following decimal numbers as binary numbers: (a) 45 (b) 128 (c) 30 7

8 Number Systems of Arbitrary Base So far we have seen two different number systems of different bases: decimal (base 10) and binary (base 2). Now lets think about the general idea of a number system with base B. What was special about the numbers 10 and 2 in the two number systems we studied? There were a couple of important things that I hope you noticed: The place values were powers of the base. The digits were allowed to be integers from 0 to the base minus 1. So just by changing the base of a number system, we can change the way in which numbers are represented completely. The base characterizes the number system. This motivates the following definition: A base B number system is a number system in which every place value corresponds to a power of B and a digit may only be an integer from 0 to (B 1). For example, consider the base 6 number system. Place values correspond to powers of 6 and digits may be integers from 0 to (6 1) = 5. An example of a base 6 number is It has the expanded form below and can be written in a place value chart as follows: (5)(6 5 ) + (4)(6 4 ) + (0)(6 3 ) + (1)(6 2 ) + (2)(6 1 ) + (3)(6 0 ) Place Value Place Value Digit

9 Restrictions on Choice of Base We cannot have 0 or 1 as a base. The main problem is that there is no meaning to place value for these bases because their powers are equal to themselves regardless of the exponent (and even undefined in the case of 0 0 ). Can you think of another reason why we can t have 0 or 1 as a base? A base must be a positive integer because we count using positive integers. A place value must be countable. A base B number system with B > 10 can be created with appropriate numerals. In the B 0 place value, we would count 1, 2, 3,..., 8, 9,..., B 1. So if B = 12 let s say, we would have to put 10 and 11 in the B 0 place value (ie. treat them as one digit). But they are two-digit numbers! However, if we use a to represent 10 and b to represent 11, then numbers like 22 or 23 could be written in base 12 as 1a and 1b respectively. The reason we have trouble creating a base B number system with B > 10 is that we use Arabic numerals for our numbers. These numerals were created for a base 10 system. Other ancient civilizations used base B number systems with B > 10, such as the Mayans. To make such a system work, they used different numerals then we do. In general, if you define your own set of numerals, there are infinitely many base B number systems that you could make. Think about why we chose to use base

10 Problem Set *difficult **challenge 1. Write (1)(2 6 ) + (4)(2 2 ) + (1)(2 1 ) as a binary number. 2. Write (1)(2 4 ) + (1)(2 2 ) + (1)(2 0 ) as a decimal number. 3. Write (5)(8 5 ) + (3)(8 4 ) + (7)(8 1 ) as a base 8 number. 4. Convert the base 3 number to a decimal number. 5. Binary numbers can also be used to encrypt messages! Figure out the following secret message by converting each binary number to base 10 and then matching the base 10 number to the corresponding letter of the alphabet (ie. 3 C, 13 M, 1 A) *6. A palindrome is a positive integer whose digits are the same when read forwards or backwards. For example, 2002 is a palindrome. How many more 3-digit palindromes exist in the decimal number system than in the binary number system? 7. The Mayans used a base 20 number system. How do you think they learned to count? What number system do you think we would use if we learned to count with only our fingers and not our thumbs? 8. Consider the base 5 number system. (a) What are the place values in this number system? (b) What integers are allowed to be digits in this number system? (c) Write the following decimal numbers as base 5 numbers: i. 700 ii. 127 iii Invent your own set of numerals and make a base B number system with B > Let s add binary numbers! Do the following binary calculations by first converting the numbers to base 10, performing the calculations, and then converting the answer back to binary. (a) (b) (c) (d) (e)

11 *11. Use the results from the previous question to do the following binary calculations without converting to base 10. Check your answers by converting the numbers to base 10 and then performing the calculations. Hint: Think of place value and carrying (as with base 10 addition). (a) (b) (c) (d) (e) *12. (a) How many different 5-digit binary numbers are there? (b) How many different 5-digit binary numbers are there that have 1 as the last digit? (c) How many different 5-digit base B numbers are there for a given B? Assume 2 B 10. **13. For a given positive integer n 2, (a) How many different n-digit binary numbers are there? (b) How many different n-digit binary numbers are there that have 1 as the last digit? (c) How many different n-digit base B numbers are there for a given B? Assume 2 B 10. **14. A friend gives you a challenge: There are seven stacks of coins that look the same. Each stack has exactly 100 coins. There are two stacks that have counterfeit coins, and all 100 coins in each of those two stacks are counterfeit. Your task is to figure out which two of the seven stacks contain the counterfeit coins. The counterfeit coins weigh 11g each, while the real coins weigh 10g each. You have an electric balance, but you can only use it to make one measurement. How can you determine which two stacks contain the counterfeit coins with only one use of the balance? Explain why the strategy works. Hint: Think about taking different numbers of coins from each of the stacks and placing them on the balance together. Think about the important numbers in the binary number system. 11

12 Answers I (a) i. (1)(10 5 ) + (2)(10 3 ) + (7)(10 1 ) + (9)(10 0 ) ii. (5)(10 2 ) + (9)(10 1 ) + (3)(10 0 ) iii. (5)(10 4 ) + (4)(10 3 ) + (1)(10 0 ) (b) i ii iii II (a) < < (b) i. 51 ii. 15 iii. 512 III (a) (b) (c) Problem Set

13 5. MATH IS FUN The Mayans likely used their fingers, thumbs, and toes to count. We would likely count in the base 8 number system if we learned to count with only using our fingers (no thumbs). 8. (a) Powers of 5 (b) A digit can be any one of the integers 0, 1, 2, 3, 4. (c) i ii iii Answers may vary. 10. (a) 0 (b) 1 (c) 1 (d) 10 (e) (a) 11 (b) 1111 (c) 100 (d) 1000 (e) (a) 16 (b) 8 (c) B 5 B (a) 2 n 2 n 1 ( ) 1 (b) (2 n 2 n 1 ) 2 (c) B n B n See solutions. 13

### Addition Methods. Methods Jottings Expanded Compact Examples 8 + 7 = 15

Addition Methods Methods Jottings Expanded Compact Examples 8 + 7 = 15 48 + 36 = 84 or: Write the numbers in columns. Adding the tens first: 47 + 76 110 13 123 Adding the units first: 47 + 76 13 110 123

### Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

### Number Systems and Radix Conversion

Number Systems and Radix Conversion Sanjay Rajopadhye, Colorado State University 1 Introduction These notes for CS 270 describe polynomial number systems. The material is not in the textbook, but will

### Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

### Grade 7/8 Math Circles Sequences and Series

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Sequences and Series November 30, 2012 What are sequences? A sequence is an ordered

### Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Lights, Camera, Primes! Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions Today, we re going

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### Math Circle Beginners Group October 18, 2015

Math Circle Beginners Group October 18, 2015 Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd

### Grade 7/8 Math Circles February 3/4, 2015 Arithmetic Aerobics

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Mental Math is Good For You! Grade 7/8 Math Circles February 3/4, 2015 Arithmetic Aerobics You ve probably

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### MATHS ACTIVITIES FOR REGISTRATION TIME

MATHS ACTIVITIES FOR REGISTRATION TIME At the beginning of the year, pair children as partners. You could match different ability children for support. Target Number Write a target number on the board.

### Section 1.4 Place Value Systems of Numeration in Other Bases

Section.4 Place Value Systems of Numeration in Other Bases Other Bases The Hindu-Arabic system that is used in most of the world today is a positional value system with a base of ten. The simplest reason

### 6 3 4 9 = 6 10 + 3 10 + 4 10 + 9 10

Lesson The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base- system. When you

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### Unit 7 The Number System: Multiplying and Dividing Integers

Unit 7 The Number System: Multiplying and Dividing Integers Introduction In this unit, students will multiply and divide integers, and multiply positive and negative fractions by integers. Students will

### Sunny Hills Math Club Decimal Numbers Lesson 4

Are you tired of finding common denominators to add fractions? Are you tired of converting mixed fractions into improper fractions, just to multiply and convert them back? Are you tired of reducing fractions

### Binary Number System. 16. Binary Numbers. Base 10 digits: 0 1 2 3 4 5 6 7 8 9. Base 2 digits: 0 1

Binary Number System 1 Base 10 digits: 0 1 2 3 4 5 6 7 8 9 Base 2 digits: 0 1 Recall that in base 10, the digits of a number are just coefficients of powers of the base (10): 417 = 4 * 10 2 + 1 * 10 1

### NUMBER SYSTEMS. William Stallings

NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html

### Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:

Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules

### PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

### CISC - Curriculum & Instruction Steering Committee. California County Superintendents Educational Services Association

CISC - Curriculum & Instruction Steering Committee California County Superintendents Educational Services Association Primary Content Module IV The Winning EQUATION NUMBER SENSE: Factors of Whole Numbers

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### Pre-Algebra Lecture 6

Pre-Algebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals

### for the Bill Hanlon bill@hanlonmath.com

Strategies for Learning the Math Facts Bill Hanlon bill@hanlonmath.com The more sophisticated mental operations in mathematics of analysis, synthesis, and evaluation are impossible without rapid and accurate

### Lecture 2. Binary and Hexadecimal Numbers

Lecture 2 Binary and Hexadecimal Numbers Purpose: Review binary and hexadecimal number representations Convert directly from one base to another base Review addition and subtraction in binary representations

### Training Manual. Pre-Employment Math. Version 1.1

Training Manual Pre-Employment Math Version 1.1 Created April 2012 1 Table of Contents Item # Training Topic Page # 1. Operations with Whole Numbers... 3 2. Operations with Decimal Numbers... 4 3. Operations

### 6 The Hindu-Arabic System (800 BC)

6 The Hindu-Arabic System (800 BC) Today the most universally used system of numeration is the Hindu-Arabic system, also known as the decimal system or base ten system. The system was named for the Indian

### Decomposing Numbers (Operations and Algebraic Thinking)

Decomposing Numbers (Operations and Algebraic Thinking) Kindergarten Formative Assessment Lesson Designed and revised by Kentucky Department of Education Mathematics Specialists Field-tested by Kentucky

### Decimals and other fractions

Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very

### Paramedic Program Pre-Admission Mathematics Test Study Guide

Paramedic Program Pre-Admission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page

### Decimal Notations for Fractions Number and Operations Fractions /4.NF

Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.

### The Euclidean Algorithm

The Euclidean Algorithm A METHOD FOR FINDING THE GREATEST COMMON DIVISOR FOR TWO LARGE NUMBERS To be successful using this method you have got to know how to divide. If this is something that you have

### A Short Guide to Significant Figures

A Short Guide to Significant Figures Quick Reference Section Here are the basic rules for significant figures - read the full text of this guide to gain a complete understanding of what these rules really

### Grade 7 & 8 Math Circles October 19, 2011 Prime Numbers

1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7 & 8 Math Circles October 19, 2011 Prime Numbers Factors Definition: A factor of a number is a whole

### 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal:

Exercises 1 - number representations Questions 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal: (a) 3012 (b) - 435 2. For each of

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### Grade 7/8 Math Circles Fall 2012 Factors and Primes

1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Factors and Primes Factors Definition: A factor of a number is a whole

### THE BINARY NUMBER SYSTEM

THE BINARY NUMBER SYSTEM Dr. Robert P. Webber, Longwood University Our civilization uses the base 10 or decimal place value system. Each digit in a number represents a power of 10. For example, 365.42

### Tom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.

Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find

### Charlesworth School Year Group Maths Targets

Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve

### Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### Base Conversion written by Cathy Saxton

Base Conversion written by Cathy Saxton 1. Base 10 In base 10, the digits, from right to left, specify the 1 s, 10 s, 100 s, 1000 s, etc. These are powers of 10 (10 x ): 10 0 = 1, 10 1 = 10, 10 2 = 100,

### Fractions to decimals

Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of

### Preliminary Mathematics

Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### Linear Equations and Inequalities

Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................

### ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite

ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,

### Using Patterns of Integer Exponents

8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. How can you develop and use the properties of integer exponents? The table below shows powers of

### The gas can has a capacity of 4.17 gallons and weighs 3.4 pounds.

hundred million\$ ten------ million\$ million\$ 00,000,000 0,000,000,000,000 00,000 0,000,000 00 0 0 0 0 0 0 0 0 0 Session 26 Decimal Fractions Explain the meaning of the values stated in the following sentence.

### WSMA Decimal Numbers Lesson 4

Thousands Hundreds Tens Ones Decimal Tenths Hundredths Thousandths WSMA Decimal Numbers Lesson 4 Are you tired of finding common denominators to add fractions? Are you tired of converting mixed fractions

### MATHEMATICS. Y5 Multiplication and Division 5330 Square numbers, prime numbers, factors and multiples. Equipment. MathSphere

MATHEMATICS Y5 Multiplication and Division 5330 Square numbers, prime numbers, factors and multiples Paper, pencil, ruler. Equipment MathSphere 5330 Square numbers, prime numbers, factors and multiples

### Day One: Least Common Multiple

Grade Level/Course: 5 th /6 th Grade Math Lesson/Unit Plan Name: Using Prime Factors to find LCM and GCF. Rationale/Lesson Abstract: The objective of this two- part lesson is to give students a clear understanding

### Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern.

INTEGERS Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern. Like all number sets, integers were invented to describe

### No Solution Equations Let s look at the following equation: 2 +3=2 +7

5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

### Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman

Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman hundredths tenths ones tens Decimal Art An Introduction to Decimals Directions: Part 1: Coloring Have children

### Accuplacer Arithmetic Study Guide

Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how

### SAT Math Strategies Quiz

When you are stumped on an SAT or ACT math question, there are two very useful strategies that may help you to get the correct answer: 1) work with the answers; and 2) plug in real numbers. This review

### Exponents. Exponents tell us how many times to multiply a base number by itself.

Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,

### Lecture 1. Basic Concepts of Set Theory, Functions and Relations

September 7, 2005 p. 1 Lecture 1. Basic Concepts of Set Theory, Functions and Relations 0. Preliminaries...1 1. Basic Concepts of Set Theory...1 1.1. Sets and elements...1 1.2. Specification of sets...2

### Scientific Notation. Section 7-1 Part 2

Scientific Notation Section 7-1 Part 2 Goals Goal To write numbers in scientific notation and standard form. To compare and order numbers using scientific notation. Vocabulary Scientific Notation Powers

### Count the Dots Binary Numbers

Activity 1 Count the Dots Binary Numbers Summary Data in computers is stored and transmitted as a series of zeros and ones. How can we represent words and numbers using just these two symbols? Curriculum

### Math Review. Numbers. Place Value. Rounding Whole Numbers. Place value thousands hundreds tens ones

Math Review Knowing basic math concepts and knowing when to apply them are essential skills. You should know how to add, subtract, multiply, divide, calculate percentages, and manipulate fractions. This

### What Is Singapore Math?

What Is Singapore Math? You may be wondering what Singapore Math is all about, and with good reason. This is a totally new kind of math for you and your child. What you may not know is that Singapore has

### Using Proportions to Solve Percent Problems I

RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### Number of Divisors. Terms. Factors, prime factorization, exponents, Materials. Transparencies Activity Sheets Calculators

of Divisors Purpose: Participants will investigate the relationship between the prime-factored form of a number and its total number of factors. Overview: In small groups, participants will generate the

### Oct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8

ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: -Our standard number system is base, also

### Introduction to Fractions, Equivalent and Simplifying (1-2 days)

Introduction to Fractions, Equivalent and Simplifying (1-2 days) 1. Fraction 2. Numerator 3. Denominator 4. Equivalent 5. Simplest form Real World Examples: 1. Fractions in general, why and where we use

### Grade 5 Mathematics Curriculum Guideline Scott Foresman - Addison Wesley 2008. Chapter 1: Place, Value, Adding, and Subtracting

Grade 5 Math Pacing Guide Page 1 of 9 Grade 5 Mathematics Curriculum Guideline Scott Foresman - Addison Wesley 2008 Test Preparation Timeline Recommendation: September - November Chapters 1-5 December

### Maths Workshop for Parents 2. Fractions and Algebra

Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)

### Today. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real- Time Systems: Binary Arithmetic

Today Binary addition Representing negative numbers 2 Binary Addition Consider the following binary numbers: 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 How do we add these numbers? 3 Binary Addition 0 0 1 0 0 1 1

### The Babylonian Number System

By Mark Swanson The Babylonian Number System Essential Questions: Why did the Babylonians use a number system based on 60 instead of 10? How did the Babylonian number system work? How was the Base 60 number

### Unit 6 Number and Operations in Base Ten: Decimals

Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### Integers, I, is a set of numbers that include positive and negative numbers and zero.

Grade 9 Math Unit 3: Rational Numbers Section 3.1: What is a Rational Number? Integers, I, is a set of numbers that include positive and negative numbers and zero. Imagine a number line These numbers are

### Positional Numbering System

APPENDIX B Positional Numbering System A positional numbering system uses a set of symbols. The value that each symbol represents, however, depends on its face value and its place value, the value associated

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Indicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities.

3 rd Grade Math Learning Targets Algebra: Indicator 1: Use procedures to transform algebraic expressions. 3.A.1.1. Students are able to explain the relationship between repeated addition and multiplication.

### Grade 6 Mathematics Performance Level Descriptors

Limited Grade 6 Mathematics Performance Level Descriptors A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Grade 6 Mathematics. A student at this

### NUMBER SYSTEMS APPENDIX D. You will learn about the following in this appendix:

APPENDIX D NUMBER SYSTEMS You will learn about the following in this appendix: The four important number systems in computing binary, octal, decimal, and hexadecimal. A number system converter program

### Number Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi)

Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi) INTRODUCTION System- A number system defines a set of values to represent quantity. We talk about the number of people

### Computer Science 281 Binary and Hexadecimal Review

Computer Science 281 Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### 4.3 TABLE 3 TABLE 4. 1342 five 1 125 3 25 4 5 2 1 125 75 20 2 222.

.3 Conversion Between Number Bases 169.3 Conversion Between Number Bases Although the numeration systems discussed in the opening section were all base ten, other bases have occurred historically. For

### 26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

### Playing with Numbers

PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also

Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of

### Digital System Design Prof. D Roychoudhry Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur

Digital System Design Prof. D Roychoudhry Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture - 04 Digital Logic II May, I before starting the today s lecture

### CSI 333 Lecture 1 Number Systems

CSI 333 Lecture 1 Number Systems 1 1 / 23 Basics of Number Systems Ref: Appendix C of Deitel & Deitel. Weighted Positional Notation: 192 = 2 10 0 + 9 10 1 + 1 10 2 General: Digit sequence : d n 1 d n 2...

### 1 ST GRADE COMMON CORE STANDARDS FOR SAXON MATH

1 ST GRADE COMMON CORE STANDARDS FOR SAXON MATH Calendar The following tables show the CCSS focus of The Meeting activities, which appear at the beginning of each numbered lesson and are taught daily,

### Binary Representation

Binary Representation The basis of all digital data is binary representation. Binary - means two 1, 0 True, False Hot, Cold On, Off We must tbe able to handle more than just values for real world problems

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### The string of digits 101101 in the binary number system represents the quantity

Data Representation Section 3.1 Data Types Registers contain either data or control information Control information is a bit or group of bits used to specify the sequence of command signals needed for

### Review of Scientific Notation and Significant Figures

II-1 Scientific Notation Review of Scientific Notation and Significant Figures Frequently numbers that occur in physics and other sciences are either very large or very small. For example, the speed of

### Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions

Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in

### Negative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2

4 (4-) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a -year

### 25 Integers: Addition and Subtraction

25 Integers: Addition and Subtraction Whole numbers and their operations were developed as a direct result of people s need to count. But nowadays many quantitative needs aside from counting require numbers