presentation on number system

The Real Number System

A Math 10C Collection of Graphical Interpretations

The math equivalent of interpretive dancing?

Brooke and Molly

Whole numbers

Rational numbers

Irrational numbers

Imaginary numbers

5 i 2 = -25

Real numbers

Natural numbers

Real Numbers ex: 2, sqrt(3), 7/22, -45/46, -9

Rational Numbers ex: 6/7, -4.56, -5/7, 4500009234

Integers ex: -4, 98989, 67, -6, -9000

Whole Numbers

Ex: 0, 2, 5, 88

Natural Numbers

Ex: 1, 2, 6, 8, 99

ex: sqrt(2), π

Ex: sqrt(-2),

J U L I A S ’ S L I D E

Jacob ’ s Slide

Fynn and Emma

Real Number System

Real numbers can be represented on a number line (includes irrational and rational numbers.)

Rational numbers can be written as a ratio/fraction.

Irrational numbers cannot be expressed as a ratio/fraction because their decimal form does not end nor is it a repetitive pattern.

Natural numbers are positive integers otherwise known as non-negative numbers that are not a fraction.

Whole numbers are “counting” numbers including 0.

Integers include all “Whole numbers” as well as their negative counterparts.

Imaginary numbers are numbers made by combining an imaginary unit (i) with a real number. They are a negative real number’s square root.

Elizabeth & Erin

~narcissism~

I dabble Sage/Clayton

♡ Isie and Lily’s slide♡ (✿●︿●)

Claire and Liza

Natural: Whole numbers that are positive. Not including zero.

Whole: Positive whole numbers. Including zero.

Integers: All whole numbers, including negatives and zero.

Rational: number with an end, and/or whose next number is predictable.

Irrational: A number with no definitive end, and whose next number cannot be predicted.

Real: A number with a value that can be put on a number line.

Imaginary: The square root of a negative number.

Victoria and Sarah

Real Number: Any number that has a value and can be put on the number line.

Irrational Number: A number that does that does not have a definitive end and the next digit is not predictive.

Rational Number: A number that has a definitive end or the next digit is predictive.

Integer Number: A number excluding fractions and decimals and can be negative, positive, and zero.

Whole: A number excluding decimals and fractions, positive, including zero.

Natural: Positive whole numbers not including zero.

Real Numbers

Irrational Number

Imaginary Numbers

Imaginary Numbers: A number that has a negative square root represented by i

Imaginary Irrational Numbers: A number that does that does not have a definitive end and the next digit is not predictive and has a negative square root

Imaginary Rational: A number that has a definitive end or the next digit is predictive and has a negative square root

Imaginary Integers: A number excluding fractions and decimals and can be negative, positive, and zero and has a negative square root

Imaginary Whole Numbers: A number excluding decimals and fractions, positive, including zero and has a negative square root

Imaginary Natural: Positive whole numbers not including zero.

i Irrational

Imaginary #s

Rational #s

Positive Whole #s above and including 1

All positive #s with no decimals

All whole rational #s without decimals

All #s that have repeating or ending decimals

Square root of a negative and 0

By: Sebastien and Matthew

All numbers except Imaginary numbers

Irrational #s

A # with an infinite # of decimals that never repeat

√ of a negative and 0

Irrational Numbers

Numbers with an infinite number of decimals that never repeat

By Julian & Sasha

Kyra and Danica Real Number System

Numbers which cannot be represented as a fraction. Numbers with an infinite decimal are irrational.

1, 2, 3, 4, 5, 6

0, 1, 2, 3, 4, 5

Any negative or positive number without a decimal

…-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...

Any number which can be represented as a fraction. Decimals which have an end, or have a repeating group are rational.

4.2, 73.823954,

An imaginary number is any i, 3 i, -7i, 21.4372 i, π i, 0 (both real and imaginary).

Real number multiplied by

the imaginary unit i, where i² = -1

Number System Presentation

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Number Systems and Operations - Mathematics - 6th Grade

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• Math Article

Number System

The number system or the numeral system is the system of naming or representing numbers. We know that a number is a mathematical value that helps to count or measure objects and it helps in performing various mathematical calculations. There are different types of number systems in Maths like decimal number system, binary number system, octal number system, and hexadecimal number system. In this article, we are going to learn what is a number system in Maths, different types, and conversion procedures with many number system examples in detail .  Also, check mathematics for grade 12 here.

What is Number System in Maths?

A number system is defined as a system of writing to express numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction, multiplication and division.

The value of any digit in a number can be determined by:

• Its position in the number
• The base of the number system

Before discussing the different types of number system examples, first, let us discuss what is a number?

What is a Number?

A number is a mathematical value used for counting or measuring or labelling objects. Numbers are used to performing arithmetic calculations.  Examples of numbers are natural numbers, whole numbers, rational and irrational numbers, etc. 0 is also a number that represents a null value. 

A number has many other variations such as even and odd numbers, prime and composite numbers. Even and odd terms are used when a number is divisible by 2 or not, whereas prime and composite differentiate between the numbers that have only two factors and more than two factors, respectively.

In a number system, these numbers are used as digits. 0 and 1 are the most common digits in the number system, that are used to represent binary numbers. On the other hand, 0 to 9 digits are also used for other number systems. Let us learn here the types of number systems.

Types of Number Systems

There are various types of number systems in mathematics. The four most common number system types are:

• Decimal number system (Base- 10)
• Binary number system (Base- 2)
• Octal number system (Base-8)
• Hexadecimal number system (Base- 16)

Now, let us discuss the different types of number systems with examples.

Decimal Number System (Base 10 Number System)

The decimal number system has a base of 10 because it uses ten digits from 0 to 9. In the decimal number system, the positions successive to the left of the decimal point represent units, tens, hundreds, thousands and so on. This system is expressed in decimal numbers . Every position shows a particular power of the base (10).

Example of Decimal Number System:

The decimal number 1457 consists of the digit 7 in the units position, 5 in the tens place, 4 in the hundreds position, and 1 in the thousands place whose value can be written as:

(1×10 3 ) + (4×10 2 ) + (5×10 1 ) + (7×10 0 )

(1×1000) + (4×100) + (5×10) + (7×1)

1000 + 400 + 50 + 7

Binary Number System (Base 2 Number System)

The base 2 number system is also known as the Binary number system wherein, only two binary digits exist, i.e., 0 and 1. Specifically, the usual base-2 is a radix of 2. The figures described under this system are known as binary numbers which are the combination of 0 and 1. For example, 110101 is a binary number.

We can convert any system into binary and vice versa.

Write (14) 10 as a binary number.

Base 2 Number System Example

∴ (14) 10 = 1110 2

Octal Number System (Base 8 Number System)

In the octal number system , the base is 8 and it uses numbers from 0 to 7 to represent numbers. Octal numbers are commonly used in computer applications. Converting an octal number to decimal is the same as decimal conversion and is explained below using an example.

Example: Convert 215 8 into decimal.

215 8 = 2 × 8 2 + 1 × 8 1 + 5 × 8 0

= 2 × 64 + 1 × 8 + 5 × 1

= 128 + 8 + 5

Hexadecimal Number System (Base 16 Number System)

In the hexadecimal system, numbers are written or represented with base 16. In the hexadecimal system, the numbers are first represented just like in the decimal system, i.e. from 0 to 9. Then, the numbers are represented using the alphabet from A to F. The below-given table shows the representation of numbers in the hexadecimal number system .

Number System Chart

In the number system chart, the base values and the digits of different number systems can be found. Below is the chart of the numeral system.

Number System Conversion

Numbers can be represented in any of the number system categories like binary, decimal, hexadecimal, etc. Also, any number which is represented in any of the number system types can be easily converted to another. Check the detailed lesson on the conversions of number systems to learn how to convert numbers in decimal to binary and vice versa, hexadecimal to binary and vice versa, and octal to binary and vice versa using various examples.

With the help of the different conversion procedures explained above, now let us discuss in brief about the conversion of one number system to the other number system by taking a random number.

Assume the number 349. Thus, the number 349 in different number systems is as follows:

The number 349 in the binary number system is 101011101

The number 349 in the decimal number system is 349.

The number 349 in the octal number system is 535.

The number 349 in the hexadecimal number system is 15D

Number System Solved Examples

Convert (1056) 16 to an octal number.

Given, 1056 16 is a hex number.

First we need to convert the given hexadecimal number into decimal number

= 1 × 16 3 + 0 × 16 2 + 5 × 16 1 + 6 × 16 0

= 4096 + 0 + 80 + 6

= (4182) 10

Now we will convert this decimal number to the required octal number by repetitively dividing by 8.

Therefore, taking the value of the remainder from bottom to top, we get;

(4182) 10 = (10126) 8

Therefore, 

(1056) 16 = (10126) 8

Convert (1001001100) 2 to a decimal number.

(1001001100) 2

= 1 × 2 9 + 0 × 2 8 + 0 × 2 7 + 1 × 2 6 + 0 × 2 5 + 0 × 2 4 + 1 × 2 3 + 1 × 2 2 + 0 × 2 1 + 0 × 2 0

= 512 + 64 + 8 + 4

Convert 10101 2 into an octal number.

 10101 2 is the binary number

We can write the given binary number as,

Now as we know, in the octal number system,

Therefore, the required octal number is (25) 8

Convert hexadecimal 2C to decimal number.

We need to convert 2C 16  into binary numbers first.

2C → 00101100

Now convert 00101100 2 into a decimal number.

101100 = 1 × 2 5  + 0 × 2 4 + 1 × 2 3  + 1 × 2 2  + 0 × 2 1 + 0 × 2 0

= 32 + 8 + 4

Video Lesson on Numeral System

Number System Questions

• Convert (242) 10 into hexadecimal. [ Answer: (F2) 16 ]
• Convert 0.52 into an octal number. [ Answer: 4121]
• Subtract 1101 2 and 1010 2 . [ Answer: 0010]
• Represent 5C6 in decimal. [ Answer:  1478]
• Represent binary number 1.1 in decimal. [ Answer: 1.5]

Also Check: Binary Operations

Computer Numeral System (Number System in Computers)

When we type any letter or word, the computer translates them into numbers since computers can understand only numbers. A computer can understand only a few symbols called digits and these symbols describe different values depending on the position they hold in the number. In general, the binary number system is used in computers. However, the octal, decimal and hexadecimal systems are also used sometimes.

More Topics Related to Number Systems

Frequently Asked Questions on Number System

What is number system and its types.

The number system is simply a system to represent or express numbers. There are various types of number systems and the most commonly used ones are decimal number system, binary number system, octal number system, and hexadecimal number system.

Why is the Number System Important?

The number system helps to represent numbers in a small symbol set. Computers, in general, use binary numbers 0 and 1 to keep the calculations simple and to keep the amount of necessary circuitry less, which results in the least amount of space, energy consumption and cost.

What is Base 1 Number System Called?

The base 1 number system is called the unary numeral system and is the simplest numeral system to represent natural numbers.

What is the equivalent binary number for the decimal number 43?

How to convert 30 8 into a decimal number.

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Number System Conversions

Number system conversions lecture l2.2 section 2.3 number system conversions hex, binary, and octal to decimal binary hex binary octal hex octal ... – powerpoint ppt presentation.

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Number Systems

Dec 20, 2019

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Number Systems. CNS 3320 – Numerical Software Engineering. Fixed-size Number Systems. Fixed-point vs. Floating-point Fixed point systems fix the maximum number of places before and after the decimal Integers are a fixed-point system with 0 decimals Advantage of Fixed-point systems?

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Number Systems CNS 3320 – Numerical Software Engineering

Fixed-size Number Systems • Fixed-point vs. Floating-point • Fixed point systems fix the maximum number of places before and after the decimal • Integers are a fixed-point system with 0 decimals • Advantage of Fixed-point systems? • Disadvantage?

Advantages of fixed-point • They are evenly spaced within their range of values • Floating-point numbers are not! • So they behave like integers • Operations are truncated to a fixed decimal point • Additions and subtractions within range are exact • No memory is wasted storing exponents • You get dependable accuracy • Moderate absolute error only in last digit (* and /) • Uniform throughout the entire system

Disadvantages of fixed-point • In a word: range • There aren’t enough numbers covered • The values needed in scientific computation typically cover a range beyond what is feasible to store in a fixed-point machine word or double-word • But we’d still like to fit numbers into machine units like fixed-point systems can • Like words and double-words (registers) • For speed

Fixed-point Example • Consider the following fixed-point number system, F1: • base = 10 • precision = 4 • decimals = 1 • F1 has 19,999 evenly-spaced numbers: • {-999.9, -999.8, …, 0, … 999.8, 999.9} • How many bits are needed to store such a number? • 1 for sign • 16 for mantissa (because each digit requires 4 bits) • Note: we don’t convert the entire number to binary, just a digit at a time (BCD) • We’re using base 10, not 2! • 17 total (more efficient encodings exist)

Types of ErrorTake Two • Absolute vs. relative • Absolute = |x – y| • Relative = |x – y| / |x| • Percentage error (preferred) • Consider the relative error of representing x = 865.54 in F1: • (865.54 – 865.5) / 865.54 = .00005 • Now, how about .86554: • (.86554 - .9) / .86554 = .04 • The relative error depends on the number of significant digits we can have, which depends on the magnitude of the number using fixed-point • Bummer

Floating-point Number Systems • Use “scientific notation” • They store a “significand” (aka “mantissa” or “coefficient”) of a fixed number of digits, along with an exponent and a sign • The number of digits stored does not depend on the magnitude of the number • You merely adjust the exponent • Which “floats” the decimal

A Toy Floating-point System • Consider the floating-point system F2: • base = 10 • precision = 4 • exponent range [-2, 3] • This system represents numbers of the form:

The Numbers in F2 • A sample: • 9999 (= 9.999 x 103) (largest magnitude) • - 80.12 • .0001 • .00002 (= 0.002 x 10-2) • 0 • 0.001 x 10-2 = .00001 (smallest positive magnitude)

The Numbers in F2 • Are not evenly spaced • Why? • How many numbers are there? • We can’t tell easily right now, but an upper bound is: • 104 x 6 x 2 = 120,000 (It’s actually 109,999) • How many bits are necessary to store such numbers? • 1 for sign, 3 for exponent (0->5 maps to -2->3), 16 for mantissa (4 x 4) • 20 total

Storage Efficiency • With 17 bits we can store 19,999 fixed-point numbers • Approx. 1,141 numbers per bit • With 20 bits we can store 109,999 floating-point numbers • Approx. 5,500 numbers per bit • Almost a 5-fold increase (4.82)!

Rounding Error in F2 • The absolute error depends on the exponent • Because the numbers aren’t evenly spaced • Consider the relative error in approximating 865.54, then .86554: • (865.54 – 865.5) / 865.54 = .00005 • (.86554 – .8655) / .86554 = .00005 • Depends only on digits, not the magnitude

Different Bases • Consider representing .1 in a base-2 system • 1/10 = 1/10102 • Use long division : .000110011001100…1010 | 1.000000000000000 • 1/10 is an infinite (repeating) decimal in base 2! • This is why 1 - .2 - .2 - .2 - .2 - .2 != 0 in a binary floating-point system

Formal Definition of FP Systems • A Floating-number systems is the set of numbers defined by the following integral parameters: • A base, B • A precision, p • A minimum exponent, m (usually negative) • A maximum exponent, M

Unnormalized FP Systems • Numbers of the form:d0.d1d2d3…dp-1 x Bewhere 0 <= di < B for all the iand m <= e <= M • Not all such numbers are unique • We’ll overcome that problem

A Sample FP System • Consider F3: • B = 2 • P = 3 • m = -1, M = 1 • List the numbers of F3 • What is the cardinality of F3? • What are the different spacings between the numbers of F3?

The Numbers of F38 bit patterns – only 16 unique numbers

The Problem with Unnormalized Systems • There are multiple ways to represent the same number • 0.1 x 2 == 1.0 x 2-1 • This leads to implementation inefficiencies • Difficult to compare numbers • Inefficient algorithms for floating-point arithmetic • Different bit patterns yield same results

Normalized Systems • Require that d0 not be zero • Solves the duplicate problem • But other problems arise • The number 0 is not representable! • We’ll solve this later • Added bonus for binary • The leading digit must be 1 • So we won’t store it! We’ll just assume it’s there • This increases the cardinality of the system vs. unnormalized

A Normalized FP System • F4 (same parameters as F3) • B = 2 • p = 3 (but it will logically be 4) • m = -1, M = 1 • If we explicitly store d0, we only get 24 distinct numbers • Because the first bit must be 1, leaving 2 bits free • But we will assume d0 = 1 • And not store it! (Only works for base = 2) • Giving 4 bits altogether (the first being 1)

The Numbers of F48 bit patterns – 24 unique numbers (but different range vs. F3)

Properties of FP Systems • Consider the system (B, p, m, M) • Numbers are of the form: • d0.d1d2…dp-1 x Be, m <= e <= M, d0 > 0 • What is the spacing between adjacent numbers? • It is the value contributed by the last digit: • 0.00…1 x Be = B1-p x Be = B1-p+e • This is B1-p for the interval [1.0, B1) • Increases going right; decreases going left

Relative Spacing in FP Systems • As we mentioned before, it’s fairly uniform throughout the system • Consider the range [Be, Be+1]: • {Be, Be+B1-p+e, Be+2B1-p+e, … Be+1-B1-p+e, Be+1} • The relative spacing between adjacent numbers is: • Between B-p and B1-p (a factor of B) • Called the system “wobble” • The second reason why 2 is the best base for FP systems! • It’s the smallest possible wobble • Independent of e!

Machine Epsilon • A measure of the “granularity” of a FP system • Upper bound of relative spacing (which affects relative roundoff error) of all consecutive numbers • We just computed this: ε = B1-p • It is also the spacing between 1.0 and its neighbor the right (see next slide) • We will use ε to tune our algorithms to the FP system being used • We can’t require smaller relative errors than ε • See epsilon.cpp

Computing Machine Parameters • They’re already available via <limits> • But they didn’t used to be • And you may not be using C/C++ forever • It is possible to determine by programming what B, p, and ε are! • See parms2.cpp

The “Most Important Fact” About Floating-point Numbers • Recall that the spacing between numbers in [Be, Be+1] is B1-p+e = B1-pBe = εBe • If |x| is in [Be, Be+1], then Be <= |x| <= Be+1=> spacing = εBe <= ε|x| <= εBe+1=> εBe-1 <= ε|x|/B <= εBe = spacing=> ε|x|/B <= spacing at x <= ε|x| • The last line is the fact to remember • We’ll use it in designing algorithms

Error in Floating-point Computations • Due to the fixed size of the FP system • Roundoff error occurs because the true answer of a computation may not be in the FP system • Cancellation in subtraction is also nasty problem • Errors can propagate through a sequence of operations • May actually increase or decrease

Measuring Roundoff Error • A single FP computation may result in a number between two consecutive FP numbers • The FP number returned depends on the Rounding Mode • Round to nearest (the most accurate) • Round down (toward negative infinity) • Round up (toward positive infinity) • Round toward zero

Measuring Roundoff Error(continued) • The absolute error of a FP computation is at least the size of the interval between adjacent numbers • aka “one unit in the last place” • Abbreviated as “ulp” • ulp(x) denotes the spacing of the current interval • We already derived this • ulp(x) = B1-p+e = B1-pBe = εBe • We already observed that the relative spacing is fairly uniform throughout the FP system • Within the system “wobble” • With larger numbers, the absolute error will, alas, be larger • Dem’s da breaks

Measuring Roundoff Error(continued) • Sometimes, instead of relative error, we’ll ask, “by how many ulps do two numbers differ?” • Same as asking: “How many floating-point intervals are there between the two numbers” • If we’re only off by a few ulps (intervals), we’re happy • ulps(x,y) is defined as the number of floating-point intervals between numbers • If the numbers have different signs, or if either is 0, then ulps(x,y) is ∞

ulps(x, y) • Recall F4: • B = 10, p = 4, m = -2, M = 3 • Calculate ulps(.99985, 1.0013) • These numbers bracket the following consecutive numbers of F4: • .9999, 1.000, 1.001 • Giving two complete intervals + two partial intervals = .5 + 1 + 1 + .3 = 2.8 ulps • In program 1 we will approximate this • We will get either 2 or 3, depending on how the actual numbers round

Example of Tuning an Algorithm • Suppose someone writes a root-finding routine using the bisection method: • Start with 2 x-values, a and b, that bracket a root • i.e., f(a) and f(b) have different signs • Replace a or b by the midpoint of [a,b] • So that the new f(a) and f(b) still have different signs • Stop when |b – a| < some input tolerance • See bisect1.cpp

The Problem • The input tolerance may be unrealistically small • It may be smaller than the spacing between adjacent floating-point numbers in the neighborhood of the solution • Endless loop! • Solution: • Reset tolerance to max(tol, ε|a|, ε|b|) • Represents the spacing between adjacent numbers in the neighborhood of the solution (see bisect2.cpp) • Often we’ll use relative error instead • Bound it by ε

Cancellation • Occurs when subtracting two nearly equal numbers • The leading digits will be identical • They cancel each other out (subtract to 0) • Most of the significant digits can be lost • Subsequent computations have large errors • Because the roundoff has been promoted to a more significant digit position • The problem with the quadratic example • Because b and sqrt(b2-4ac) were very close • Sometimes b2 and 4ac can be close, too • Not much we can do about that (use even higher precision, if possible, or try case-by-case tricks)

Differing Magnitudes • When very large and very small numbers combine • Sometimes not a problem • Smaller numbers are ignored (treated as 0) • Fine if the number is growing • But consider the exp(-x) case • Initial terms in the Taylor series are large • Their natural roundoff (in their last digit) is in a higher-valued digit than the final true answer • All digits are bad! • Made a difference because we were subtracting • The running sum was ultimately decreasing

Potential Overflow • Adding two numbers can result in overflow • IEEE systems have a way of “handling” this • But it’s best to avoid it • Example: (a + b) / 2 in bisection • Numerator can overflow! • Alternative: a + (b-a)/2 • Also checking f(a)*f(c) < 0 can overflow • Try f(a)/fabs(f(a))*f(c), or write a sign function

Error Analysis • We know that the floating-point approximation to a number x has relative error < ε • Rewrite this as:

Error in Adding 2 Numbers • For simplicity, we’ll assume the relative roundoff error of each single operation is the same δ (they’re all bounded by ε anyway):

Now compute the relative error: So the error of the sum is roughly the sum of the errors (2δ) plus a hair, but the two errors could be nice and offset each other a little.

Now consider the sum x1 + x2 + x3: • We’ll even ignore the initial errors in approximating the original numbers • Let’s just see what addition itself does when repeated • We’ll again call each relative error δ The smaller x1 and x2 are the better. Rule of thumb: add smallest to largest when possible.

Error Propagation Example • The mathematical nature of a formula can cause error to grow or to diminish • It’s important to examine how errors may propagate in iterative calculations • Example:

Integrating by parts, we end up with a recurrence relation:

The initial error in E1 (call it δ) gets magnified • By a factor of n! (n-factorial)

Solution • Rewrite the recurrence backwards, and use an initial guess for En • The initial guess doesn’t have to be too close, as you’ll see (analysis on next slide) • See en.cpp

The initial error, δ, gets dampened with each iteration.

Summary • Floating-point is better than fixed-point for: • Range of available numbers • Storage efficiency • Bounded relative error • Floating-point is less resource intensive than using arbitrary precision algorithms • Floating-point is subject to roundoff error • Because the set of numbers is finite • The absolute error grows with the magnitude • Because numbers aren’t evenly spaced (gap widens) • But the relative error stays bounded • Within the system wobble

Summary • Normalized systems are preferred over unnormalized • Unique bit patterns for distinct numbers simplifies algorithms • Formulas that describe the behavior of the system are easier to derive • Storage optimization with binary • Machine epsilon is the fundamental measure of a FP system • Upper bound on relative roundoff error • Used to tune algorithms

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