Calculator used with kind permission of DoctorProAudio.com
Fill out a gray box above and click the respective 'calculate' bar under it. p-p = peak to peak.
The reference voltage for 0 dBu is 0.775 volt (0.77459667 V) and for 0 dBV it is exactly 1.0 volt.
Scroll down to find the formulas for voltage and power and the calculation of the absolute level.
The origin of the index of dBu comes from "u = unloaded" and dBV comes from "V = 1 volt". Some say:
The "u" in dBu implies that the load impedance is unspecified, unterminated, and is likely to be high.
What is dBu? A logarithmic voltage ratio with a reference voltage of V0 = 0.7746 volt ≡ 0 dBu
What is dBV? A logarithmic voltage ratio with a reference voltage of V0 = 1.0000 volt ≡ 0 dBV
The home recording level (consumer audio) of −10 dBV means 0.3162 volts, that is −7.78 dBu.
The studio recording level (pro audio) of +4 dBu means a voltage of 1.228 volts.
The maximum undistorted level of audio amplifiers is +18 dBu. In USA it is +24 dBu.
Domestic gear with a −10 dBV level is usually unbalanced. Studio gear with a +4 dBu level is always balanced. 0 VU = +4 dBu.
Scale: Level in dBu and dBV in comparison to the voltage in V
|Studio level international||+4 ●||1.228||+1.78|
|Standard level 1 Volt||+2.22||1||0 ref.|
|Standard level 0.775 Volt||0 ref.||0.775||−2.22|
|Domestic level||−7.78||0.316||−10 ●|
|The level difference between +4 dBu studio level
and −10 dBV consumer level is Δ L = 11.78 dB (12 dB).
The level difference between dBu level and dBV level is Δ L = 2.2 dB.
0 dBV equals 2.2 dBu or 0 dBu equals −2.2 dBV.
The conversion from level L(dBu) to voltage (volt) is V = 0.775 × 10(L/20).
The conversion from voltage V (volt) to level (dBu) is L = 20 × log (V/0.775).
|All field quantities, like voltage or sound pressure are
always true RMS values, if not otherwise stated.
In mathematics, the root mean square (abbreviated RMS
or rms), also known as the quadratic mean, is a statistical
measure of the magnitude of a varying quantity.
|For sinusodial voltages or currents with ohm's loads
calculations can made easier with RMS = amplitude / √2
Note - Comparing dBSPL and dBA: There is no conversion formula for
measured dBA values to sound pressure level dBSPL or vice versa.
Also you cannot convert "dBA to volts" and vice versa.
Conversion is only possible for measuring one single frequency.
Pro audio equipment often lists an A-weighted noise spec – not
because it correlates well with our hearing – but because it can
"hide" nasty hum components that make for bad noise specs.
Words to bright minds: Always wonder what a manufacturer
is hiding when they use A-weighting. *)
|We don't use the dBm in audio engineering. That belongs to power, we don't need here.
If you really need dBm with "m = 1 milliwatt or one thousandth (10−3) of a watt", go to:
The dBm calculator and power matching or impedance matching
Explanation: What is "dBFS"? (Digital Audio)
dBFS - Digital recording level
Analog levels and digital levels are different realms.
♦ An often posted question: "Please, can you help me convert from dBFS to dBu".
Never express analog signal levels in terms of dBFS.
Follow this and you will not confuse anyone.
Notice - Comparing dBu and dBFS: There is really no fixed
world standard like e.g. −20 dBFS = +4 dBu = 0dBVU.
The digital peak scale is not equivalent to the analog RMS scale.
dBu is volts - you measure it with a volt meter.
Analog audio: positive and negative voltage.
dBFS is in contrast a binary number.
Digital audio: zeroes and ones.
There is no such thing as peak volts dBu *)
It is incorrect to state peak voltage levels in dBu.
dBFS must have a minus sign at the beginning. There is not something like +6 dBFS.
There is no such standardized reference. x dBFS is a digital voltage level
(peak) and y dBVU or dBu is an analog voltage level (RMS).
Digital and analogue are two totally different realms.
That's why there is no relation between dBFS and dBVU or dBu, whatsoever.
Analog meter (ppm): attack time 10 to 300 ms − reading rms values.
Digital meter: attack time < 1 ms − reading peak values. That is really some
Advice: Watch only your digital meters and go up to 0 dBFS, but never go
over this mark. We use "headroom" in the analog domain. That is OK, but
we don't need digital "headroom" as an always "unused" forbidden
You are free to choose your private headroom, if you like that, but there is no
standard that you have to do that.
The demand for a high modulation level, stand in the contrary to the claim,
to avoid overloading.
Never take the following funny guessing game for granted. Use it only as a rough guide:
European & UK calibration for Post & Film is −18 dBFS = 0 VU = +4 dBu
BBC spec: −18 dBFS = PPM "4" = 0 dBu
American Post: −20 dBFS = 0 VU = +4 dBu
Orchestral −18 dBFS = 0 VU = +4 dBu
Rock and / or Radio −16, or −14, or −12 dBFS = 0 VU = +4 dBu
Digi 002 is only capable of −14 dBFS.
German ARD & studio PPM +6 dBu = −10 (−9) dBFS. +16 (+15) dBu = 0 dBFS. No VU.
• EBU R68-2000 - The European Broadcasting Union recommends: digital level
−9 dBFs (maximum). You have to keep the upper 9 dBs empty without any use.
The reference level is −18 dBFs. 0 dBFs is equal to +15 dBu.
Notice: 0 dBFS is the permitted maximum digital level.
The EBU broadcasters have a problem, because they want to use the old slow meters with the
dBu scale (attack 10 ms, quasi-peak) of the analog times for digital recordings.
The rest of the world looks always at the fast digital meters (attack < 1 ms, peak) with the dBFS
scale. Forget looking at the dBu scale of the meters.
It seems to come a change from QPPM-modulation to loudness (ITU/EBU) and true-peak.
Look at: EBU R 128.
|Note: The guidelines of the EBU to set the maximum gain
to −9 dB dBFS should not apply if not working for the
European Broadcasting Union. Whose maximum levels
of digital CD masters are −9dBFS, should not be
surprised if the CDs are not loud enough.
9 dB up to the top are left free with really no use.
| More readable papers of the topic "Loudness and Level":
10 things you need to know about ... EBU R 128 - the EBU loudness recommendation
Florian Camerer: Loudness On the way to nirvana - audio levelling with EBU R 128
|LUFS = Loudness Units relative to Full Scale|
The formulas for voltage and power
and the calculation of the absolute level
|To use the calculator, simply enter a value.
The calculator works in both directions of the ↔ sign.
dBm indicates that the reference power is P0 = 1 milliwatt = 0.001 watt ≡ 0 dB
Conversion of voltage or power ratios to decibels dB - table and chart
Table of Sound Pressure Levels and Corresponding Sound Pressure and Sound Intensity
RMS voltage, peak voltage and peak-to-peak voltage
The parameters of the mains or "power" sine wave form are summarized at the table below:
|Average voltage||RMS voltage (VRMS)||Peak voltage (Vp) = (Û)||Peak-to-peak voltage (Vpp)|
|0 volts||117 volts = VRMS = ~V||165 volts = √2×VRMS = 0,5 × Vpp||330 volts = 2×√2×VRMS = 2 × Vp|
|0 volts||230 volts = VRMS = ~V||325 volts = √2×VRMS = 0,5 × Vpp||650 volts = 2×√2×VRMS = 2 × Vp|
|The value VRMS of an alternating voltage V (t) = V0 × f(t)is defined so that the
effective DC power corresponds VRMS2 / R = VRMS × IRMS to an ohmic resistance
of the middle resistive power of this AC voltage to the same resistance.
The crest factor means the ratio of the peak voltage to the RMS voltage.
If you need to calculate an attenuator (attenuation calculation) you calculate a voltage divider.
|Voltage||VRMS = ~V||Vp||Vpp|
|Average voltage RMS VRMS =||−||0.7071 × Vp||0.3535 × Vpp|
|Peak voltage Vp =||1.414 × VRMS||−||0.5000 × Vpp|
|Peak-to-peak voltage Vpp =||2.828 × VRMS||2.000 × Vp||−|
Different voltage levels
|Level||Level L in dB||Voltage RMS||Voltage peak-to-peak|
|European studio level - ARD broadcast level||+6 dBu||1.55 V||4.38 V|
|International studio level - USA||+4 dBu||1.228 V||3.47 V|
|Domestic recording (Consumer units)||−10 dBV||0.3162 V ≡ −7.78 dBu||0.894 V|
|Sound pressure level (auditory threshold)||0 dB||2×10−5 Pa ≡ 0 dBSPL||5.66×10−5 Pa|
|Reference studio level re 0.775 volt||0 dBu||0.7746 V||2.19 V|
|Reference studio level re 1 volt||0 dBV||1.0000 V||2.828 V|
International reference values
|Physical unit||Level unit||Reference value||Note|
|Voltage||V0 = 0.775 V||≡ 0 dBu||Audio engineering, no impedance reference!|
|Voltage||V0 = 1 V||≡ 0 dBV||Audio engineering, USA|
|Voltage||V0 = 1×10−6 V||HF receiver and amplifier technology|
|Voltage||V0 = 0.224 V||HF technology - Reference 1 mW at Z = 50 Ω|
|Voltage||V = 1.228 V||Studio level +4 dBu, USA - Reference 0.7746 V|
|Voltage||V = 1.55 V||Studio level +6 dBu, ARD - Reference 0.7746 V|
|Voltage||V = 0.3162 V||Home recording level −10 dBV - Reference 1.0 V ≡ −7.78 dBu|
|Sound pressure||p0 = 2×10−5 Pa||≡ 0 dB||Sound Pressure Level SPL, auditory threshold (Sound field size)|
|Sound particle velocity||v0 = 5×10−8 m/s||≡ 0 dB|
|Sound intensity||I0 = 1×10−12 W/m2||≡ 0 dB||Threshold of pain at 1 W/m2 (Sound energy size)|
|Power||P0 = 1 W||≡ 0 dBW||The reference impedance must always be told|
|Power||P0 = 1 mW||≡ 0 dBm||Z = 600 Ω (telephones) or Z = 50 Ω (antennas)|
|Electric field strength||E0 = 1×10−6 V/m|
|Decibels (dB) Calculator
Decibels are defined as ten times the log of a power ratio. Decibels convert
multiplication and division calculations into simple addition and subtraction operations.
This calculator converts between decibels, voltage gain (or current), and power gain.
Just fill in one field and the calculator will convert the other two fields.
Equations: Level in dB: L = 20 × log (V1/V2) = 10 × log (P1/P2)
|The dBm is a logarithmic measure of power compared to 1 mW,
that means it is power dependent.
It can be converted to a voltage, if the load impedance is known.
Typically the impedance (load) is 600 ohms.
Equation: Level in dBm: LP = 10 × log (P / 0.001)
|Simple rule of thumb: When working with power: 3 dB is twice, 10 dB is 10 times.
When working with voltage or current: 6 dB is twice, 20 dB is 10 times.
|Why is the bandwidth and the cutoff frequency found at the level of "−3 dB"?
Why we always take 3 dB down gain of a filter?
Full width at half maximum (FWHM).
Answer: That is the point where the energy (power) is fallen to the value ½ or 0.5 = 50 percent of the initial power as energy
quantity, that is equivalent to (−)3 dB = 10×log(0.5). A (−)3 dB power drop is a decrease of 50 % to the value of 50%.
There the voltage is fallen to the value of √(½) or 0.7071 = 70.71 percent of the initial voltage as field quantity equivalent to
(−)3 dB = 20×log(0.7071). A (−)3 dB voltage drop is a decrease of 29.29 % to the value of 70.71 %.
|(−)3 dB implies ½ the electric power and since the power is proportional to the
square of voltage, the value will be 0.7071 or 70.71 % of the passband voltage.
√½ = 1/√2 = √0,5 = 0,7071. P ~ V2, that is 0,5 ~ 0,70712.
|Sound engineers and sound designers ("ear people") mostly use the usual (sound) field quantity. That'swhy they say:
The cutoff frequency of a device (microphone, amplifier, loudspeaker) is the frequency at which the output voltage level
is decreased to a value of (−)3 dB below the input voltage level (0 dB).
● (−)3 dB corresponds to a factor of √½ = 1/√2 = 0.7071, which is 70.71% of the input voltage.
Acousticians and sound protectors ("noise fighters") seem to like more the (sound) energy quantity. They tell us:
The cutoff frequency of a device (microphone, amplifier, loudspeaker) is the frequency at which the output power level is
decreased to a value of (−)3 dB below the input power level (0 dB).
● (−)3 dB corresponds to a factor of ½ = 0.5, which is 50% of the input power (half the value)..
|Note: Power gain (power amplification) is not common in audio engineering.
Even power amplifiers for loudspeakers don't amplify the power.
They amplify the audio voltage that moves the voice coil.
|Sound field quantities
Sound pressure, sound or particle velocity,
particle displacement or displacement amplitude,
(voltage, current, electric resistance).
Inverse Distance Law 1/r
|Sound energy quantities
Sound intensity, sound energy density,
sound energy, acoustic power.
Inverse Square Law 1/r²
|Note: A sound field quantity (sound pressure p, electric voltage V) is not a sound energy
quantity (sound intensity I, sound power Pak). I ~ p2 or P ~ V2. Sometimes you can hear
the statement: The cutoff frequency is there where the level L is decreased by (−)3 dB.
Whatever the user wants to tell us so accurately: Level is level or dB is dB.


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...As it turned out, teaching people to move better was far more difficult than teaching physics! In fact, I would say that it was in the comparison of these two fields that I came to better understand how to effectively communicate ideas, i.e. to teach. In fact, my favorite part of teaching is wh...
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The Department of Mathematics and Statistical Science offers three graduate degrees in mathematics (M.A.T., M.S., and Ph.D.) and one graduate degree in statistical science (M.S.), along with two graduate certificates.
Graduate Degrees in Mathematics
An advanced degree in mathematics prepares students for a career in teaching, research, and various technical fields.
Master of Arts in Teaching Mathematics
The M.A.T. degree is designed primarily for certified teachers who wish to strengthen their subject matter preparation. We offer this degree on campus and through distance learning. Note: Receiving this degree does not give certification in high school teaching.
Master of Science in Mathematics
The M.S. degree prepares students to enter a Ph.D. program, to teach at some colleges, and for many positions in business and industry. We offer a non-thesis and thesis option for our M.S. program, both of which can be completed in two years.
Doctor of Philosophy in Mathematics
A Ph.D. in mathematics gives you the opportunity to teach mathematics at the university level, with the opportunity for original research.
Graduate Degree in Statistical Science
Master of Science in Statistical Science
The objective of the M.S. program in Statistical Science is to provide sound training in the fundamental principles and techniques of statistics. Graduates will be equipped for a variety of statistical careers in industry, business, agriculture, government and biomedical fields or to engage in further study in the doctoral level. This degree is also available online.
There is a thesis option and a non-thesis internship option, to provide flexibility in meeting our students academic and career goals.
The Department of Mathematics and Statistical Science offers two graduate certificates: Data Science Certificate and Statistics Certificate.
Bioinformatics and Computational Biology
The University of Idaho also has strong departments of biology, computer science, and several engineering disciplines. Several of our faculty members and graduate students participate in our interdisciplinary graduate program in Bioinformatics and Computational Biology.


If you need to replace a broken domestic light bulb you will need to take note of the two sets of figures printed on the glass. One of the figures is likely to be 240 V. This number confirms that for the bulb to work correctly it should be connected to the UK mains supply. The second set of figures could be 10 W or 60 W or 100 W, or similar. But what do these figures mean? If you bought one of each of these bulbs and connected them to the same supply the difference between them would be obvious. The 10 W (ten watt) bulb would be quite dim, the 60 W bulb would be brighter, and the 100 W bulb would glow even brighter.
Bulbs are energy converters. They convert electrical energy into heat and light energy. The numbers on the bulbs tell us how quickly they do this. The 100 W bulb is very bright as it is converting 100 J (joules) of electrical energy into 100 J of heat and light energy every second it is turned on. The 60 W bulb is less bright as it is converting 60 J of electrical energy into 60 J of heat and light energy every second. The 10 W bulb is dim as it is converting just 10 J of electrical energy into 10 J of heat and light energy every second.
The rate at which a device converts energy is called the power or power rating of the device. For example, the brightest bulb above has a power rating of 100 W because it converts energy at the rate of 100 J every second. The dullest bulb has a power rating of 10 W because it converts energy at the rate of 10 J every second.
power and energy It follows from the above that we can work out how much energy a device has converted by using the equation:
energy converted = power in watts × time in seconds
or E = P × t
The energy calculated is measured in joules.
example Calculate how much energy is used when a 600 W TV set is turned on for 30 minutes, using E = P × t.
E = 600 × (30 × 60)
E = 1,080,000 J or 1,080 kJ
The energy used in joules when a TV set is turned on for just 30 minutes is quite a large number. If we measure the total amount of electricity we use in the home over three months in joules the number would be incredibly large and clumsy. (A little bit like asking you how far it is to the town centre from where you live but asking for the distance in centimetres!)
To get around this problem we usually measure domestic electrical energy in much bigger units, called the kilowatt-hour or unit.
Imagine a 1 kW fire turned on for one hour – it will use 1 kWh (kilowatt-hour) or 1 unit of energy. If it is turned on for two hours it will use 2 kWh or 2 units of energy. If a 2 kW fire is turned on for three hours it will use 6 kWh or six units of energy, and so on.
Therefore we can calculate the energy used in kWh or units by any device, using the equation:
number of units used = power of device in kW × time in hours
To work out the total cost of the electricity we have used, we now simply multiply the number of units used by the cost of one unit.
example Calculate the cost of having a 3 kW electric fire turned on for one full day if the cost of one unit is 9 p, using E = P × t.
E = 3 × 24
E = 72 units
total cost = number of units × cost per unit
total cost = 72 × 9
total cost = 648 p or £6.48


Water flows over a section of Niagara Falls at a rate of 1.2 × 106 kg/s and falls 71 m. The acceleration of gravity is 9.8 m/s2 . How many 70 W bulbs can be lit with this power?
I understand that the rate of flow of water was meant to be 1.2 x 10^6 kg/s rather than 1.2 x 106 kg/s. The question is solves with this rate.
As the water comes down the fall, its potential energy is converted into kinetic energy that may be used for generating electric power.
If we assume that this whole energy is converted into electricity with 100 percent efficiency, the electric energy generated is same as potential energy of water converted into kinetic energy. The rate of generation of this electric energy ids given by the formula:
Power (in watts) = Energy (in joule)/s.
Energy/s = m*g*h
Where m = mass of water per second = 1.2 x 10^6 kg/s
g = acceleration due to gravity = 9.8 m/s^2, and
h = Height of fall = 71 m.
Therefore electric power generated
= m*g*h = (1.2 x 10^6)*9.8*71 = 834.96 x 10^6 W
number of 70 watts bulb that can be lit by this power is given by the formula:
Number of bulbs = (Total power generated)/(Power used per bulb)
= (834.96 x 10^6)/70 = 11 982 000 bulbs.
Answer: 11 982 000 bulbs can be lit.
The mass, m of water falling in one second =m=1.2*106kg=127.2kg.
The acceleration of falling = g=9.8m/s^2.
The force generated by water due to falling = mg Newton =127.2*9.8 N=1246.56Newton.
The work done by the force = force* distance = force * height of fall=1246.56*71m=88505.76 watts
This energy is equal to (88505.76/70)=1264.368 or 1264.368 times the energy required of a 70 w bulb.


Moment of Inertia and its significance :
According to newton's first law of motion the body continues to stay in the state of rest or in the state of motion. This property of inertia possessed by all the matter is the measures of mass in translatory motion. Similarly when a body is rotating or fixed about an axis. Any change in its state of rest or motion can be brought about by a couple or Torque the greater the opposition to change greater is the inertia. It is this rotational inertia of the body is called its moment of inertia about its axis of rotation.
The analogy between the two types of motion is the mass defined as " Coefficient of inertia for translatory motion" and the moment of inertia defined as "Coefficient of rotational inertia in rotating motion". In translatory motion the motion of the body depends on the mass alone.
In rotatory motion, the moment of inertia of the body depends not only on the mass of the body but also upon the effective distance of the body from the axis of rotation. The effective distance K of the particles from the axis of rotation is called the radius of gyration about that axis and is equal to root mean square distance of the particles from the axis.
"The radius of gyration of a body about a given axis of rotation is the distance from the axis of rotation is the distance from the axis at which if the whole mass of the body were to be concentrated, the moment of inertia of the body about the axis of rotation would be the same as with the actual distribution of mass.
The mass of body remains the same, irrespective of rotation, the value of radius of gyration about the axis depends upon.
a) The position and direction of the axis of rotation
b) The distribution of the mass of the body about this axis, the value for the same body is different at different inclination.
The radius of gyration of a body about a given axis of rotation given an inclination of the distribution of the mass of the body about it and hence the effect of distribution mass on the moment of inertia about that axis.
|Guest Author: pretty 25 Aug 2014|
|plz upload proper significance of moment of inertia|
|Guest Author: kishore ram Dhaka 16 Jan 2015|
|moment of inertia is always use in the case of rotation of a body about an axis and if moment of inertia high then you have to apply high torque to rotate the object and if less than less torque needed. As in linear motion if mass of body is high then high force is needed to accelerate the object.|
|Guest Author: Niku 01 May 2015|
|Nice article - I like it very much.|


This study examines the relationship between portfolios and regressions, which is desirable for educational, mathematical, and theoretical reasons. Educationally, understanding this relationship simplifies the teaching and learning of both procedures. Mathematically, portfolio optimization and regression systems are abstractly, algebraically, topologically, and structurally equivalent. One is obtained from the other as if modeling clay, without tears or discontinuities, and what one learns in one system can be applied to the other. We show portfolios and regressions are equivalent at a theoretical level as well. In the economic-financial context, this theoretical equivalence means that mean-variance, efficient portfolios are in fact optimal predictors, which is necessary for arbitrage-based investment valuation and for the study of arbitrage-based market adjustment. We use linear algebra and study the characteristics of Lagrange methods to make our point. We also provide specialized procedures to facilitate portfolio optimizations.
M. Tarrazo, Portfolios and Regressions, Journal of Financial Education, 35, (2009), 56-74.


NC NAEP MATH SCORES
SHOW STATE'S STUDENTS ABOVE THE
NATIONAL AVERAGE IN MATHEMATICS, AT THE
NATIONAL AVERAGE IN READING
North Carolina fourth and eighth grade students are performing at the national average or better, according to the 2011 National Assessment of Educational Progress (NAEP) results just released by the National Center for Educational Statistics (NCES). The NAEP is now required of every state and is often referred to as the nation's Report Card.
"I am committed to ensuring that all of our students perform at grade level, particularly in reading, writing and mathematics," Gov. Bev Perdue said. "These test scores show that we have made real progress, but still have more work to do to meet my goal of ensuring that every child has a chance to succeed in a career, in a two- or four-year college or in technical training. We continue to move kids forward in North Carolina. Our high school graduation rate is higher than ever, and today's scores in math and reading are going up as we invest in Career and College: Ready, Set, Go!"
State Superintendent June Atkinson noted that North Carolina public schools have been very focused on improving instruction and learning for students over the past decade and are making additional improvements to continue this focus.
"Mathematics and reading are critical skills for all students and provide the foundation for learning in other disciplines," said State Superintendent June Atkinson. "Our NAEP scores show that we have made progress in mathematics, but we continue to face challenges in improving reading performance.
"Our state's move next year to the Common Core state standards in reading and mathematics will give us the tools to help more students move into the Proficient and Advanced achievement levels."
North Carolina's reading scores show North Carolina fourth grade students earning an average score of 221 while the nation is at 220. At eighth grade, North Carolina's average score is 263, and the nation's average is 264. In reading, North Carolina's average score is not considered to be statistically different from the national scores, according to the NCES.
In mathematics, North Carolina fourth graders posted an average score of 245 as compared to the national average of 240. For eighth graders, the average mathematics score was 286, while the national average was 283. In mathematics, North Carolina's average score is considered to be statistically higher than the national scores, according to NCES.
In the recent years of NAEP, North Carolina students have shown steady improvement in mathematics. Reading performance has experienced more fluctuations.
Compared to other states, North Carolina's average fourth grade mathematics score was higher than 30 other states or jurisdictions, lower than 6 others and in the same performance category as 15 other states or jurisdictions. Eighth grade mathematics scores placed North Carolina lower than 11 other states or jurisdictions, higher than 26 others and the same as 14 others.
On reading, North Carolina fourth graders were lower than 13 states or jurisdictions, higher than 15 states or jurisdictions and not significantly different from 23 others. In eighth grade, North Carolina's reading scores placed the state below 27 other states and jurisdictions, higher than 11 states or jurisdictions and the same as 13 others.
In addition to providing the average scores for each grade and subject, the NAEP performance also is reported by the percentage of students scoring in three achievement levels: Basic, Proficient and Advanced. In fourth grade mathematics, North Carolina has 44 percent of fourth graders scoring at the Basic level; 38 percent at Proficient and 7 percent at Advanced. When the state began participating in NAEP in 1992, approximately half of all fourth graders were below Basic in mathematics. In 2011, 12 percent of North Carolina fourth graders scored below Basic. The percentages scoring in each category have fluctuated from one testing year to the next, but the trend in fourth grade mathematics is toward having more students at Basic or above.
At the fourth grade level in mathematics, the nation has 18 percent below Basic; 42 percent at Basic, 33 percent at Proficient and 6 percent at Advanced.
For eighth grade mathematics, North Carolina has 38 percent of its students at Basic; 27 percent at Proficient; and 10 percent at Advanced. Nationally, those numbers are 39 percent at Basic, 26 percent Proficient and 8 percent Advanced. When North Carolina began participating in the NAEP mathematics assessment in 1990, 62 percent of students were scoring below the Basic level. This has declined to 25 percent.
In reading, North Carolina's scores and the nation's scores have remained more consistent. In 2011, North Carolina posted 34 percent of its fourth grade students at Basic, 26 percent at Proficient and 8 percent at Advanced. Nationally, the averages in fourth grade reading were 34 percent at Basic, 25 percent at Proficient and 7 percent at Advanced. In 1992 when the NAEP began assessing reading, North Carolina's average score was significantly lower and the percentage of students who were below Basic was 44 percent versus 32 percent in 2011, significantly.
For eighth grade reading, North Carolina scores showed 43 percent of students at the Basic level, 28 percent at Proficient and 3 percent Advanced. The nation's numbers were 43 percent at Basic, 29 percent at Proficient and 3 percent Advanced. In 1998 when eighth grade reading was first measured by the NAEP, North Carolina had 24 percent of students below Basic. In 2011 that number was 26 percent.
NAEP is given in reading and mathematics every other year. The NAEP assesses reading in both literary and informational texts. The assessments are given to a representative sample of North Carolina students selected by the NCES as representative of the state.
The NAEP assesses mathematics in five content areas: number properties and operations; measurement; geometry; data analysis, statistics, and probability; and algebra.
NAEP results for the nation reflect the performance of students attending public schools (including charter schools), private schools, the Department of Defense schools, and Bureau of Indian Education schools. Only national and state-level data are available because the NAEP is given to a sample of students in each state. The NAEP assessments are conducted in reading, mathematics, science, writing, civics, economics, geography, US history and the arts. Reading and mathematics are the two subjects assessed every two years with state-level results reported.
Comparisons between North Carolina students and the nation are made using public school scores only.
NOTE :: Various file formats are used on this page that may require download. If larger than 1mb, it will take longer to download. For instructions or more information, please visit our download page.
About the North Carolina Department of Public Instruction:
The North Carolina Department of Public Instruction provides leadership to 115 local public school districts and 160 charter schools serving over 1.5 million students in kindergarten through high school graduation. The agency is responsible for all aspects of the state's public school system and works under the direction of the North Carolina State Board of Education.
For more information:
NCDPI Communication and Information Division, 919.807.3450.


Prof. Youjin Zhang
Tsingua University, China
Special Cubic Hodge Integrals and the Fractional Volterra Hierarchy
We show that the generating function of cubic Hodge integrals satisfying the local Calabi-Yau condition is the tau function of a particular solution of an integrable hierarchy called the fractional Volterra hierarchy. This integrable hierarchy is a certain generalization of the Volterra lattice hierarchy (also called the discrete KdV hierarchy) which is well known in the theory of nonlinear integrable systems.
Thursday 21 Jul 2020, 10AM Beijing Time.


To use this converter, just pick a unit to transform from, a unit to convert to, then form the value you want to convert. The an outcome will be displayed immediately.
You are watching: How many teaspoons are in 4 ounces
This converter accepts decimal, integer and fractional values as input, therefore you can input worths like: 1, 4, 0.5, 1.9, 1/2, 3 1/2, etc.
Note that to enter a combined number favor 1 1/2, you show leave a room between the integer and the fraction.
The numerical an outcome exactness will certainly be follow to de number o far-reaching figures the you choose.
When the result shows one or more fractions, girlfriend should think about its colors according to the table below:
Exact portion or 0% 1% 2% 5%10%15%
These colors represent the maximum approximation error because that each fraction. If the error does no fit your need, you have to use the decimal value and also possibly boost the number of significant figures.
Please, if friend find any issues in this calculator, or if girlfriend have any type of suggestions, please contact us.
See more: How Many 3 Letter Combinations Are There, Three Letter Combinations
The maximum approximation error for the fractions presented in this app are according v these colors:
Exact fraction 1% 2% 5%10%15%
One to countless converters
Examples the Volume Conversions4-1/2 gallons to cubic inches981 milliliters come quarts5 quarts to ounces2 tablespoons come milliliters1 pint come cupsliters to gallons4 tablespoons come ounces3/16 gallons to ounces11/64 cubic inches to litersounces come pintscubic meters to liters1 quart come ouncesmilliliters come cupsdrops to teaspoons40000000000000 cups to pints4 ounces come millilitersounces to gallonsliters to milliliters50 ounces to cups292 drops to milliliters
This application software application is because that educational functions only. We space not responsible for any special, incidental, indirect or consequential damages of any type of kind occurring out the or in link with the usage or power of this software.


Original link: http://mingxin.life/index.php/archives/337/
I think there are two types of learning, one is general knowledge, including various common sense, life principles, etc., and the other is ability.
Leaving aside general knowledge, the purpose of the latter is to acquire a certain ability, and existing knowledge is a shortcut. But the premise is often ignored, that is, you must be able to walk by yourself. The result of putting the cart before the horse is that knowledge has become a mountain that hinders us.
When I was in school, I was very confident in learning data, why? Because I learned a formula when I was in elementary school, and deduced another formula by myself. I felt that I had made a big discovery, and I excitedly told the teacher. The teacher told me that I could learn it next year, but he still suspected that I had read the textbooks for the senior grades. I was very frustrated because the formula I discovered had been discovered by someone else, but I was also very happy because the first person who discovered it was a mathematician. If I was born earlier than him, this formula should have my name on it. Already—a childhood thought.
Of course, this incident discouraged my interest in deriving mathematical formulas, because I don’t know if the formulas I deduced have been discovered by others. But this kind of self-confidence has always been with me, and I have never doubted my ability to learn. Until high school, my mathematics, physics and chemistry have always been very good.
In my opinion, the exploration of human beings in the field of mathematics is like walking a path. When you first started on the road, you only need to do addition and subtraction, and you can get negative numbers when you do subtraction. Moving forward, you need to do multiplication and division. When you have decimals and fractions, you will find that some are indivisible, and you will have infinite decimals. Going forward, you need to calculate the area and volume, and you have the square, the cubic, the square root, and even a geometry. Going deeper, equations, mechanics, calculus and so on. These concepts or discoveries are like signposts, recording the footsteps of human exploration.
When we go to study mathematics, we are retracing this path. But the first thing we have to learn is how to walk, that is, the basic method of studying mathematics, and then we run along the road signs of our predecessors, quickly explore this road, and when we run to the end of the road, we can slowly explore it. What are the places to go. Even on the way, you can also pay attention to the open spaces beside the road without road signs.
For those who are not interested in mathematics, it is enough to learn how to calculate. More complex calculations can be learned after use, or directly use a computer.
I am not interested in pure theoretical research, I prefer to study and research purposefully. Because I believe that the actual needs of social life promote the development of science and technology, and dragon-slaying skills that have nothing to do with social life needs will disappear no matter how advanced they are, while related skills will continue to develop.
In addition to the fun of learning itself, I went to school to enter the university, and the purpose of the university was to find a job. Studying at work is because of work needs, and restarting study after the age of 30 is to solve my life problems.
There is no need or problem, and learning aimlessly is just another way to pass the time. To pass the time, we must find the way we like, just like when we get tired of playing a game, we will naturally change it.
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This site is only for collection, and the copyright belongs to the original author.


If you are obliged to follow a mandated math curriculum at your school, then I suggest you read no further because this site is probably not suitable for you. However, if you are looking for something that is different, yet might spark an interest in either you or your students, then it might be worthwhile for you to continue reading.
In 1992 I wrote an alternative curriculum for Pre Calculus students who enjoyed mathematics, but not the rigor of mathematics. I named the course Mathematical Investigations and it soon became recognized as the most enjoyable math course at our school for many of our students. One of topics that we delved into was titled Calculator Limitations and Extensions. The students’ reactions were so positive that I made a similar presentation with the same title at the Wisconsin Mathematics Council Annual Meeting to 60 math educators. Of the five problems that were examined, none could be solved directly by use of a calculator without applying the appropriate mathematical concepts. Interestingly, many of the math teachers struggled with the problems as much as my students did.
Eventually, I wrote an enrichment booklet series called Ten Plus One for American Educational Products that contained similar type problems. Of the original six booklets, four remain in their catalog for sale, but the other two have been discontinued so I am free to sell them myself through TpT. This booklet is titled as Calculator Investigations.


I have another DimeCast that went live this morning. This one is entitled “Exploring Expressions with Spark.”.
Here’s what we cover:
”We are going to take a look at the various different ways you can use Expressions with the Spark View engine. We are going to review simply scenarios such as math based expressions and move up to looking at nested if-then-else statements.”


Coder Academy at the Redford Township Library
Redford Township Library
25320 Six Mile Road
Redford Charter Township, MI 48240
Kids in grads 3-5 can participate in this coding workshop. Participant must have basic math skills, including addition and subtraction and plotting coordinates. You must bring your own laptop and headphones/earbuds (please remember to bring charger pack), a notebook, writing utensils, calculator and your own lunch.
Participant should be interested and enthusiastic about the subject material, this is the only way to get the best experience, and must be ready to participate and have a great attitude each day. Registration required as class size is limited to 18.


A new three step iterative method for solving nonlinear equations is introduced based on the following scheme: Let be an initial guess sufficiently close to a simple root of the equation . The iterative step consists two predictor steps:
and one corrector step:
. The authors show that if the function is sufficiently differentiable on an open interval which contains a single root, and if is sufficiently close to this root, then the proposed iterative algorithm has the fourth-order of convergence. Several numerical examples are given to illustrate the efficiency and performance of the new method.


M111 Math as a Human Pursuit Lab Semester 121
Cane Toad Population Growth Lab Name _____________________
Web site for Film: Canetoads the Conquest
From 1935-37, the
American marine toad -- also called the Cane Toad --(Bufo marinus) was
Year Area Occupied (square km)
1. We are going to use Logger Pro software to find the best fitting exponential curve for this toad data and then use it to predict future growth. The primary reason we are using Logger Pro is to become familiar with its interface for some experiments we will be doing.
2. First save the following “Experiment File” for Logger Pro that has the above toad data already entered in it. (Right click on this link ToadPop-121.cmbl and chose “Save Link Target As” – then save the file to the N: drive).
3. Start Logger Pro by clicking on the Logger Pro Icon (Close Tip of the Day if showing, and then Click on “Continue without interface”)
4. Choose Analyze – Curve Fit. Select Manual on the Fit Type. Click on Define Function and then press enter. Into the box labeled Define User Function type "P*(1+r)^x", enter the name as “expon” and click on OK. Then in the boxes to the right enter the values: 32800 for P and 0.05 for r. Then click on OK
5. You can see that the curve does not have a good fit since the rate .05 is not fast enough.
To delete the curve click on the x on the floating box that labels the curve.
6. Choose Analyze – Curve Fit, the function P*(1+r)^x and enter the same value for P(32800), but a different value for r. (You can use the up and down arrows by the r parameter until you think the curve is a good fit – set step size to 0.01 first). You should see a value for RMSE in the lower right of the window. The RMSE ( Root Mean Square Error) is a measure of how far away, on average, the data points are from the fitted curve. RMSE is in the units of the y-axis (toads in our case). Try to get the smallest value for RMS (root mean square error) that you can.
Write down the resulting equation that you decide upon:
7. Now choose Analyze – Curve Fit, the function P*(1+r)^x, and click on the Automatic option, then choose OK. This has Logger Pro choose the best fitting curve (-- in terms of a least squares fit -- the one that minimizes the sum of the squares of the vertical distances of the points from the curve.)
a) Write down the equation that Logger Pro chooses as the best fit for the data:
b) Using this model of exponential growth what should the population be in the year 2012 (73 years after 1939)?
(use your calculator or Maple).
c) What should the population be in the year 2039 (100 years after 1939) according to this growth pattern?
d) Check your answers to the above questions in step 6 by choosing Analyze – Interpolate. Then move the mouse to the right until you find year 2012 and read the corresponding value for the Toad Population. Write down the predicted population:
e) Repeat for the year 2039 (Elapsed time = 100). Write down the predicted population:
8. Delete the fit(s) that you made by clicking the x-button (close) on the box with the curve.
9. The data from the years 10 to 30 (1949 to 1969) seem to be linear. Try a linear fit for to this range of data:
a) (Zoom back in first). Select these data points by dragging the mouse pointer across the region from time 10 to time 30 (The region will be shaded gray) Then choose Analyze – Linear Fit.
b) Write down the equation for the line represented by this fit.
c) There are two measures displayed the goodness of linear fit RMSE and correlation.
Correlation is always between -1 and +1. Perfect positive linear correlation would have a value of 1. Perfect negative correlation would have a value of -1. Positive correlation exists between two variables if they tend to move in the same direction, negative correlation exists if they tend to move in the opposite direction (e.g. if we had a declining toad population.)
Write down the correlation shown:
Write down the RMSE shown:
d) What would the population be in the years 2012 and 2039 respectively if the growth fit this linear curve?
Check your answer by using the Analyze – Interpolate again as before.
10. To save your experiment file, choose File – Save and to exit LoggerPro, choose File – Exit.


The EDrill Math Flashcard contains five programs: Basic Addition/Subtraction Tables, Advanced Addition/Subtraction, Basic Multiplication/Division, Advanced Multiplication/Division, Advanced Time/Money. This program will take your child from basic addition/subtraction tables all the way to advanced multiplication/division. It will prompt your child with questions and check the answers for you. In addition to generating problems, all the programs keep track of correct, incorrect, and skipped problems, plus an overall average. There is also a correct and incorrect answer log that allows the parents to review the student's work.
Windows 98/Me/2000/XP, Microsoft .NET Framework 1.1 Language Pack
Limited set of math problems
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Do you have a question you would like to ask Professor Puzzler? Click here to ask your question!
Lavinia from Italy writes: "Hi professor. I have just read your post about this year's number (2017). In this post you say that we can write 2017 as a the sum of one cube plus twice another. 2017 = 113 + 2 · 73 I was wondering how you managed manage to discover this. Did you use a formula or something? Thanks, Lavinia"
Thanks for asking, Lavinia. For those who missed it, this is a reference to the following blog post: Happy New Year 2017. In this blog post I make mention of the fact Lavinia pointed out:
2017 = 113 + 2 · 73
So how did I discover this odd fact? There wasn't a formula. I found it simply by playing around with the numbers to see what interesting things I found. I don't remember exactly what I was doing when I discovered that, but it probably involved a spreadsheet. I probably was thinking, "I wonder what results I'll get if I subtract perfect cubes from 2017!"
So I probably entered the following formula into cell A1 of a spreadsheet:
= ROW()^3 (cube the row number)
And then, in B1 cell put the following formula:
= 2017 - A1
Then I would have used the "fill down" feature to populate a bunch of rows with the results of those calculations:
Incidentally, this time as I looked at the rows of numbers, I realized that there's a perfect square in the second column: 289 = 172. So we can also write:
2017 = 123 + 172
I probably did something very similar to find the perfect squares that add to 2017:
A1 = ROW()^2
B1 = 2017 - A1
C1 = SQRT(B1)
This would have given the following:
2017 = 92 + 442
Incidentally, there was another row that had no decimal, further down the spreadsheet. It was row number 44, of course!
Excel (or any other spreadsheet application) can be a great help in exploring mathematical oddities. I realize that I'm a bit spoiled, having calculators and spreadsheets to use when playing with numbers. Imagine trying to figure out quirky number facts like these without even a calculator!
Steve from Cincinnati writes: "I believe this puzzle answer is 1089.
11 X 11 = 4
22 X 22 = 16
33 X 33 = ???
"The first two equations are wrong, but they trick us into looking for a pattern rather than apply the mathematical rules...many people assume an addition function exists between each of the two pairs of integers. Not so. Therefore, thirty three multiplied by thirty three is 1089."
Hi Steve! Of course, you are 100% correct. 33 x 33 = 1089, no matter how many incorrect equations you put before it. Now that I've said that, if I stop there, I've got a really short blog post, and no one wants that - so let's fix their notation in order to turn it into a more interesting puzzle. We'll treat it as a single variable function:
Find x if:
F(11) = 4
F(22) = 16
F(33) = x
It still isn't a very good puzzle, though, because there's not enough information to come up with a single solution. In that respect, it's very similar to the "Squares Problem" I wrote up last year. Here are just three possible answers to the puzzle.
We add the digits of the number in the function's domain, and then square it. (1 + 1)2 = 4; (2 + 2)2 = 16; (3 + 3)2 = 36.
We square the number, and then add the digits: 112 = 121 and 1 + 2 + 1 = 4; 222 = 484 and 4 + 8 + 4 = 16; 332 = 1089 and 1 + 0 + 8 + 9 = 18.
We subtract 8 from the number, and then add the quotient the number and eleven: 11 - 8 + 11/11 = 4; 22 - 8 + 22/11 = 16; 33 - 8 + 33/11 = 28.
I'm sure there are other solutions; those are just the ones that popped into my head on a quick glance. Can you come up with others? Send them my way and I'll add them to this post.
The following question comes to us from fifth grader Savannah, from Ruston. "How do you convert units with exponents using the method 'king henry doesn't usually drink chocolate milk'?"
Hi Savannah. "King Henry Doesn't Usually Drink Chocolate Milk" is what we call a mnemonic; it's a phrase that's designed to help you remember something important. In this case, what we're trying to remember is the prefixes for metric units. The prefixes all begin with the same letters as the words of the mnemonic, as shown below, with an example unit:
King ⇒ kilo ⇒ kilometer
Henry ⇒ hecto ⇒ hectometer
Doesn't ⇒ deca ⇒ decameter
Usually ⇒ Unit without prefix ⇒ meter
Drink ⇒ deci ⇒ decimeter
Chocolate ⇒ centi ⇒ centimeter
Milk ⇒ milli ⇒ millimeter
So let's say I had 20 hectoliters. How would I convert that into milliliters? King Henry will help me remember the order of the units:
kilo ⇒ hecto ⇒ deca ⇒ unit (without prefix) ⇒ deci ⇒ centi ⇒ milli
If I want to go from hectoliters to milliliters, I'm moving 5 prefixes to the right, so I'm going to multiply by 105. Thus, I have 20 x 105 = 2000000 milliliters.
If I wanted to convert 5 centimeters to decameters, I am moving the opposite direction through the units, and I'm moving 3 prefixes. So this time I divide (because I'm moving in the other direction) by 103. 5/103 = 0.005 decameters.
One problem with this mnemonic is that both deca and deci begin with D, so you still have to remember which word goes with which prefix. I would try to remember that both "drink" and "deci" have the letter i in them, and that may help you out.
Good luck with your unit conversions!
"Can you explain terza rima and give an example?" ~Anon, grade 5
Terza Rima is an Italian phrase that means "third rhyme." It's a specific way of rhyming lines in a poem. I think of it as sort of a revolving door of rhymes. In each stanza of a Terza Rima poem, there are two lines that rhyme, and one line that does not. The line that doesn't rhyme provides the rhyming syllable for the next stanza. Even though it doesn't rhyme with other lines in that stanza, it provides a connection to the next stanza, thus building the whole poem into a progressive, seamless whole.
In a Terza Rima poem, the last stanza often has two rhyming lines (that's called a couplet).
In other words, the rhyme scheme looks something like this:
ABA - BCB - CDC - DD
If you wanted more than four stanzas, you could chain together as many stanzas as you want in this format.
If you have a hard time following that explanation, here's a silly poem I wrote just for you, that uses the Terza Rima rhyme scheme:
I dreamed the world was made of cookie dough.
The skies were filled with cotton candy clouds,
And from them blew a storm of whipped-cream snow.
The fields of chocolate, farmers left unplowed;
The stalks of candy-cane grew everywhere,
And gum-drops grew on bushes, low but proud.
Oh, nothing in this world seemed quite so fair
As pine tree branches bowed with sugar cones -
Enough for all the hungry crowds to share.
A whiff of spearmint on the wind was blown
O'er milk-shake streams and maple syrup lakes.
I shouted from atop my candied throne:
"This world of ours, it really takes the cake -
If it's a dream, I do not wish to wake!"
Copyright 2017 by Douglas Twitchell
Incidentally, Robert Frost wrote a terza rima sonnet titled "Acquainted with the Night." In addition, his poem "Stopping by Woods on a Snowy Evening" is not Terza Rima, but it's a very similar "chained" rhyme; each stanza has four lines. The third line doesn't rhyme with the others, but it does introduce the rhyme for the next stanza. The rhyme scheme looks like this:
AABA - BBCB - CCDC - DDDD
B.R. asks, "Can you explain what mechanical advantage is?"
Hi B.R., I'll give it a shot. Not knowing how much Physics background you have, I'll try to explain it in simple terms.
First of all, I'd like you to consider the following arrangement, which is a person pulling on a rope, which runs over a pulley, and down to an object with a weight of 400 Newtons (that's about 90 pounds, if you're used to using English units).
Now, when we talk about someone pulling on something, we're using the concept of a force. Forces are measured in Newtons (or pounds). If the object has a weight of 400 Newtons, that means the person holding the rope has to pull with a force of 400 Newtons in order to keep the object from speeding up or slowing down. If he pulled with a force greater than 400 Newtons, the object would begin accelerating upward. If he pulled with a force less than 400 Newtons, the object would begin to accelerate downward.
This system has a mechanical advantage of 1. This means that on one end, the person applies a force that is equal to the force at the other end of the system. 400 / 400 = 1.
So let's take our pulley system and make it just a little more complicated. Here I've put a second pulley at the bottom, and attached the rope to the ceiling. The object hangs from the lower pulley.
Now, one of the things that's important to understand is that if the man pulls with a force of 400 Newtons, there is tension in the rope, and the tension is the same everywhere along the rope. That tension is also measured in Newtons, and it is 400 Newtons. But here's where it gets interesting. Look at that lower pulley (that the object is hanging off). What is the tension in the rope to the left of the pulley? 400 N, right? What is the tension in the rope to the right of the pulley? It's also 400 Newtons. This gives a total of 400 + 400 = 800 Newtons. The upward force on the object is double the object's weight. This object is going to accelerate upward, even though in the first scenario, it would not have accelerated.
So if the man pulls with a force of 400 N, the output force is 800 N. The mechanical advantage is 800 / 400 = 2.
Incidentally, this means that the man could hold up an 800 Newton weight by applying a 400 Newton force. That's pretty impressive!
You could add more pulleys to the system, as shown here:
In this case, if the man is pulling with a force of 400 N, that means that there is 400 N of tension all through the rope, and so each of the lower pulleys has an upward force of 800 N, which means that the object (which is being supported from those two pulleys) has an upward force of 1600 N applied!
So the mechanical advantage is: 1600 / 400 = 4.
It seems like this is a win-win situation; you get better results by adding more and more pulleys. So why not add five hundred pulleys? If you did, you could hold up a 400 Newton object by applying a tiny, tiny force (less than 1 Newton!). So is it really a win-win situation? Not exactly. You see, there's a tradeoff for mechanical advantage. The tradeoff is the distance you pull the object. Take a look at the first diagram. If the man pulls the rope down 1 meter, the object will rise 1 meter.
But what about the second diagram? What will happen there? Well, if the man pulls the rope down 1 meter, that meter is evenly distributed on either side of the lower pulley. Which means that pulley will only rise 0.5 meters. And therefore, if he pulls down a meter, the object only rises half a meter.
What about the third diagram? In that scenario, if the man pulls the rope downward 1 meter, that meter gets evenly distributed among four sections of the rope (one on either side of the two lower pulleys). In other words, the object only moves 0.25 meters.
So, if you had 500 pulleys, you'd be in the very interesting situation that you could easily lift a 400 Newton object, but it would take you a long time to do it; for every meter you pulled the rope down, the object would only move 2 millimeters!
There are many other kinds of machines: screws, wedges, levers, axle-and-wheels, and inclined planes. For every machine, there's a tradeoff; you can make the work easier, but in doing so, you end up extending that effort over a longer distance. An inclined plane is another easy one to picture. Imagine trying to lift a grand piano 1 meter straight off the ground. Can't do it, right? But if I made a long ramp, you could push the piano up. Much easier than lifting, but the tradeoff is that you have to push it a whole lot more than 1 meter!
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Heron’s Formula Class 9 MCQ is one of the best strategies to prepare for the CBSE Class 9 Board exam. If you want to complete a grasp concept or work on one’s score, there is no method except constant practice. Students can improve their speed and accuracy by doing more Heron’s Formula class 9 MCQ which will help them all through their board test.
Heron’s Formula Class 9 MCQ Questions with Answer
Class 9 Math MCQ with answers are given here to Chapter 12 Heron’s Formula. These MCQs are based on the latest CBSE board syllabus and relate to the latest Class 9 Math syllabus. By Solving these Class 9 MCQs, you will be able to analyze all of the concepts quickly in the Chapter and get ready for the Class 9 Annual exam.
Learn Heron’s Formula Class 9 MCQ with answers pdf free download according to the latest CBSE and NCERT syllabus. Students should prepare for the examination by solving CBSE Class 9 Heron’s Formula MCQ with answers given below.
Question 1. The sides of a triangle are 56 cm, 60 cm and 52 cm long. Then the area of the triangle is
(a) 1322 cm²
(b) 1311 cm²
(c) 1344 cm²
(d) 1392 cm²
Question 2. The area of an equilateral triangle with side 2√3 cm is
(a) 5.196 cm²
(b) 0.866 cm²
(c) 3.496 cm²
(d) 1.732 cm²
Question 3. An isosceles right triangle has area 8 cm². The length of its hypotenuse is
(a) 32−−√ cm
(b) 16−−√ cm
(d) 24−−√ cm
Question 4. The perimeter of an equilateral triangle is 60 m. The area is
(a) 10√3 m²
(b) 15√3 m²
(c) 20√3 m²
(d) 100√3 m²
Question 5. The sides of a triangle are 35 cm, 54 cm and 61 cm. The length of its longest altitude is
(a) 16√5 cm
(b) 10√5 cm
(c) 24√5 cm
(d) 28 cm
Whoever needs to take the CBSE Class 9 Board Exam should look at this MCQ. To the Students who will show up in CBSE Class 9 Math Board Exams, It is suggested to practice more and more questions. Aside from the sample paper you more likely had solved. These Heron’s Formula Class 9 MCQ are ready by the subject specialists themselves.
Question 6. The length of each side of an equilateral triangle having an area of 9√3 cm² is
(a) 8 cm
(b) 36 cm
(c) 4 cm
(d) 6 cm
Question 7. If the area of an equilateral triangle is 16√3 cm², then the perimeter of the triangle is
(a) 48 cm
(b) 24 cm
(c) 12 cm
(d) 36 cm
Question 8. The base of a right triangle is 48 cm and its hypotenuse is 50 cm. The area of the triangle is
(a) 168 cm²
(b) 252 cm²
(c) 336 cm²
(d) 504 cm²
Question 9. The area of an isosceles triangle having base 2 cm and the length of one of the equal sides 4 cm is
(a) 15−−√ cm²
(b) 152−−√ cm²
(c) 215−−√5 cm²
(d) 415−−√ cm²
Question 10. The edges of a triangular board are 6 cm, 8 cm and 10 cm. The cost of painting it at the rate of 9 paise per cm² is
(a) Rs 2.00
(b) Rs 2.16
(c) Rs 2.48
(d) Rs 3.00
You can easily get good marks If you study with the help of Class 9 Heron’s Formula MCQ. We trust that information provided is useful for you. NCERT MCQ Questions for Class 9 Heron’s Formula PDF Free Download would without a doubt create positive results.
We hope the information shared above in regards to MCQ on Heron’s Formula Class 9 with Answers has been helpful to you. If you have any questions regarding CBSE Class 9 Math Solutions MCQs Pdf, write a comment below and we will get back to you as soon as possible.
Frequently Asked Question (FAQs)
How many MCQ questions are there in Class 9 Chapter 12 Math?
In Class 9 Chapter 12 Math, we have provided 10 Important MCQ Questions, But in the future, we will add more MCQs so that you can get good marks in the Class 9 exam.
Can we score good marks in Class 9 Math with the help of Heron’s Formula MCQ Questions?
Yes, MCQ Question is one of the best strategies to make your preparation better for the CBSE Board Exam. It also helps to know the student’s basic understanding of each Chapter. So, You can score good marks in the Class 9 Math exam.


Financial Planning Calculators
Technology permeates every part of modern life, but at its heart is the fact that computers are really quite good at math. Computers can crunch numbers in seconds that might stump us for hours. To prove it, look no further than online calculators. With a simple applet, you can perform calculations that would take much longer by hand. Of course, we rarely run to a calculator just to do some math for the fun of it. We’re often looking for the answer to a problem, and that’s where specialty calculators come in. When we go beyond adding and subtracting, we start to unlock the real potential of these mathematics tools. Whether you need to figure out an advanced math problem, convert between different units of measurement, get a grip on your financial situation, or figure out your retirement savings, there’s an online tool for you. You can find all manner of financial assistance tools online, and armed with these resources, anyone can get a sense of where they stand financially and how to improve their situation.
Basic Math Calculators
- Math Is Fun Calculator: Click this on-screen calculator’s buttons, or type a formula to compute your answer.
- Basic Calculator: You’ll find all of the functions you’d expect from a simple calculator in this tool.
Scientific, Visual, and Graphing Calculators
- Online Graphing Calculator: There’s no need to dig out your old graphing calculator from high school. This site brings graphing functions right to your Web browser.
- Mathway Graphing Calculator: This graphing calculator features more than a dozen functions and a visually appealing display. You can enter multiple problems, making graphing equations easy.
- Length Converter: Flipping between metric and imperial units can be bothersome, and repeated manual conversions can introduce errors. Use a converter tool (like this one for converting length) to keep your data accurate.
- Temperature Converter: It’s easy to get an approximate value when converting from Celsius to Fahrenheit (or vice versa), but an accurate temperature reading is tough to do in your head. Fortunately, tools like this one make it easy.
- Bandwidth Conversion: Every industry has its own jargon and its own need for specialized calculations. This tool makes calculating computer network bandwidth easy. It’s even handy for figuring out your home Internet speed or how long it will take to download a file.
- Currency Conversion: Traveling or doing business overseas? You’ll want to know the value of your local currency as it compares to those you’ll find abroad. This tool helps you convert between currencies using up-to-date exchange rates.
- Date Calculator: Adding or subtracting time on a calendar can be tricky. It’s more than simple math because not all months have the same number of days. This calculator makes it simple: Input your starting date and how many days, weeks, months, or years to add or subtract.
- Speed, Distance, and Time Calculator: This calculator works out the speed of a vehicle given the distance it has traveled over a set time. It can also do the inverse, finding the time or distance if you know the speed.
- Time, Speed, and Distance Calculator: This all-in-one program handles time, speed, and distance. As long as you know two of the three, this page will enable you to calculate the third.
- Mass to Weight Calculator: When calibrating instruments or performing physics simulations, it’s important to start with good data. This tool converts mass to weight, taking into account the effects of gravity and acceleration.
Financial Planning Calculators
Personal Loan Calculators
- Personal Loan Calculator: Want to know how much a personal loan will cost you every month? This is the tool for you. Input the principal, term, and interest to get the payment amount instantly.
- Simple Loan Calculator: This loan calculator helps you find your monthly payment amount. Its easy-to-use sliders are useful for what-if scenarios, so you can see the impact of varying the interest rate or length of the loan.
- Loan Calculator: This loan calculator looks simple but has a few tricks up its sleeve. It shows monthly payments and interest-only payments and also shows the total amount repaid over the life of the loan.
- Budget Calculator: We all need to start somewhere, and this basic budget tool is a great first step. Input your income and estimated expenses and you’ll instantly get a picture of your financial health.
- Income and Spending Analysis: When you’re ready to take your budgeting to the next level, this AARP calculator is a winner. It takes into account payroll taxes, spousal income, and a host of other variables.
- Monthly Budget Calculator: This monthly budget works well for an individual or a family. It includes all of the common expense types plus a few fields for those miscellaneous expenses that can be hard to categorize.
- Debt Calculator: Ever wondered how debt consolidation could impact your financial situation? Input a few data points into this tool and find out.
- Debt Consolidation Calculator: This calculator takes a bit more input, as you’ll want to include all of your debts and expenses. Once they’re entered, however, you’ll have a clear picture of how a consolidation loan could help you out.
- Amortization Loan Calculator: Knowing your monthly payment is a big help, but visually tracking the debt balance on a per-payment basis is even more powerful. That’s where this handy tool comes in.
College Financial Aid Calculators
- College Cost Calculator: Tuition can be expensive, but there are more college costs to consider. This comparison tool adds up other expenses, like room and board, books, supplies, and more. Then, it calculates how much you’ll need from grants, scholarships, and loans to cover the college experience. Fill in the data for several schools and you’ll have an easy way to compare costs.
- Loan Repayment Calculator: This loan calculator is customized for student loans, showing how much you’ll need to repay. All of the standard loan types are included, and their common interest rates are helpfully provided.
Car and Mortgage Calculators
- Auto Loan Calculator: This quick and easy tool shows you the monthly payment for an automotive loan. It’s great for determining the down payment and trade-in amount you’ll need to get to the desired payment.
- Online Loan Calculator: This advanced auto loan calculator shows the amortization of your loan as well as the total interest paid. There’s also a graph of your balance over time, which updates as you modify the loan terms.
- Payment Calculator: Go even more in depth with this auto loan payment tool. It can accommodate sales tax, variable payment frequencies, and auto rebates. It can even generate a complete amortization schedule, including depreciation.
Retirement Savings Calculator
- Retirement Income Calculator: This tool is easy to use but still gives you a quick overview of your retirement situation. You can edit the details to model multiple scenarios, such as saving more or retiring at a different age.
- Savings Calculator: Compound interest is the most powerful tool we have for building wealth. Find out how much your money could grow with this savings calculator.
- Compound Interest Calculator: This tool has the flexibility to explore multiple scenarios and to account for taxes and inflation. It can also generate a table showing how your money might grow over time.
Financial Planning Tools
- Understanding Checking Accounts: There’s no better way for young learners to get familiar with financial concepts than with hands-on roleplay. This lesson includes handouts, definitions, and interactive activities that help young people learn about checking accounts.
- Making Spending Decisions: One of the hardest financial lessons to learn is how to judge when to spend and when to save. This lesson helps younger children (pre-K through second grade) understand why finances are about choices.
- Inflation Is Real: Intended for grades 6-8, this lesson explains the concepts behind inflation, including what it is, why it matters, and how to calculate it.
- A Lifetime of Savings: Using real-world examples, this lesson guides students from grades 9-12 through the concept of compound interest.
- Teenagers and Saving: The earlier we start saving, the more time we have for compound interest to grow our wealth. Help teenagers understand saving with these tips.
- Money Management for College Students: From saving on supplies to reining in impulse buys, these tips can help any college student make the most of the money they have.
- Getting Fiscally Fit: This toolkit is intended for adults who want to get control of their finances. You might want to save up for a goal, or you might just want to know more about where you stand in your financial life.
- Mastering Your Finances: These tools from NerdWallet can help anyone take stock of their finances. It may seem like an insurmountable task, but mastering your finances can start with a few simple steps, like starting a budget and committing to saving consistently.
- Beginner’s Guide to Money Management and Budgeting: This collection of tools and tips can help adults of any age. It’s filled with helpful hints about building wealth over time, minimizing spending, and planning for retirement.
- Avoid Investment Fraud: Once you start building wealth, you’ll want to protect it. Use these tips to avoid fraud and protect your investments.
- Managing Your Money in Retirement: Once you’ve saved enough for retirement, how do you shift from saving to spending? This extensive toolkit can help make sense of retirement finances.


I want to prove double negation introduction in sequent calculus using the most basic rule set.
That is what I want to prove: from the sequent $$\Gamma \rightarrow\Phi,$$ the sequent $$\Gamma \rightarrow\neg\neg\Phi$$ is derivable.
How is this accomplished? It's seems much more difficult than in natural deduction. I think I may be able to prove the conjunction myself once I have this result.


Suppose you were preparing two-way tables of percentages for the following pairs of variables. How would you run the percentages?
a) Age and consumption of breakfast cereal.
b) Family income and confidence about the family's future.
c) Marital status and sports participation.
d) Crime rate and unemployment rate.
This solution contains an example of using a chi-squares table in MS excel as a decision making device.


Data Analysis & Regression
In this lesson we will explore data analysis issues in pharmacokinetics. In the recorded presentations we talk about the philosophy of data analysis in pharmacokinetics and pharmacodynamics and then review linear regression. Finally we describe the differences between linear regression and nonlinear regression and give a brief introduction to maximum likelihood estimation, which is the main method used in nonlinear mixed effects modelling.
You will also find a manuscript by Boxenbaum et al. This manuscript was meant to be a tutorial on pharmacokinetic data analysis and although it was published some time ago it is a useful introduction to many of the issues in nonlinear regression as applied to pharmacokinetics.
After this presentations you should have a clearer understanding of pharmacokinetic data analysis
You should understand how to select an appropriate model given a particular data set and dosing information
You should be able to diagnose model fit and perform simple statistical evaluation of the fit
1.1 Please study the video presentations below:
Please study the 4 recorded Powerpoint lecture presentations below:
You can download a PDF of the presentation files here:
In addition, study the Boxenbaum manuscript below:
You can view a pre-recorded tutorial of this lesson here. [ Video ]
N.R.Draper, H.Smith, Applied Regression Analysis, Wiley, 1998
Y. Bard , Nonlinear Parameter Estimation, Academic Press, 1974
P.L. Bonate, ‘Pharmacokinetic-Pharmacodynamic Modelling and Simulation’, Springer, 2006
M. Davidian, D.M. Giltinan, ‘Nonlinear models for repeated measures data’, Chapman & Hall, 1995


By JuR4gaN. Worksheet. At Thursday, March 26th 2020, 04:01:06 AM.
This section contains all of the graphic previews for the Differential Equations Worksheets. We have slope field, separable equation, differential equation, and exponential growth worksheets for your use. These Calculus worksheets are a good resource for students in high school.
And if you’d like an open-and-go, all-in-one resource that guides you step-by-step, check out my book: Addition Facts That Stick. It provides detailed lesson plans, fun games, and simple worksheets for every step of the process, so that you can teach your child addition facts that truly stick.
These Math Flash Cards are great for classroom practice recognizing different shapes. The set will produce thirteen different shapes. These Math Flash Cards are appropriate for Kindergarten, 1st Grade, and 2nd Grade.


We are given that `int_(0)^(1)(15xsqrt(x^2+9))dx=50sqrt(10)-135` and we are asked to evaluate `int_(1)^(0)(15usqrt(u^2+9))du` .
These are definite integrals. If we restrict the variables to the real number line, a definite integral is typically defined as the Riemann integral; i.e. an infinite sum of geometric shapes. Definite integrals can be described as a "signed" area.
By the Fundamental Theorem of Calculus, we can find the definite integral by integrating the indefinite integral and evaluating at the endpoints. `int_(a)^(b)f(x)dx=F(b)-F(a)` where F is the function whose derivative is f.
So one way to evaluate the given integral is to integrate the indefinite integral. Thus
Then noticing that d/du of u^2+9 is 2udu we can multiply inside the integral by 2 and outside by 1/2 to get:
`=15/2 * ((u^2+9)^(3/2))/(3/2)+C`
The definite integral is
But there is an easier way. There is a rule for integrals that states:
So we take the given value of the indefinite integral and multiply by negative one to get
The functions in the integrand are identical except for the variable. The only other difference between the two definite integrals is that the upper and lower limits of integration have been switched. Thus we can use the rule giving us the answer of the opposite of the original definite integral.


Photomath is an application that uses the camera of your device to read and solve mathematical problems. This application also provides the step-by-step process by which the solution is derived. As of the latest version of this app, Photomath supports basic arithmetics, fractions, decimals, linear equations, and several supplementary functions like logarithms.
|Name:||Photomath on Computer|
|Compatibility:||Windows (All Versions)|
|Security:||Checked – Virus Free|
This camera-using calculator was released in 2014 and after being praised by critics worldwide, it was also included in the Top 20 applications for learning and teaching. The Photomath application is free to download from both Play Store and App Store and can serve as a useful tool for teaching and learning mathematics. Whether it is to check homework or study for upcoming ACTs/SATs, Photomath has you covered.
What is Photomath?
The Photomath application is compatible with both Android and iOS devices where it can be used to capture and solve mathematical problems in real-time with the help of the device camera. This application makes use of the latest text recognition software, identifying and solving mathematical problems of varying complexity, both handwritten and in the printed text.
Photomath makes use of optical character recognition (OCR) technology which allows it to read the question, solve it, and provide answers within seconds. This application currently has over 100 million downloads and new mathematical functions are added to each successive version of the app.
Photomath is an extremely reliable application, explaining in a step-by-step manner how to solve tricky mathematical problems that have you stuck in your homework or assignment. Not only is this software a useful tool for students who frequently find themselves unable to solve certain questions or equations, but also helpful for teachers and parents who want to help their students or children with their schoolwork.
How to Install Photomath for PC for free?
The Photomath application has versions that are compatible with both Android and iOS and can be downloaded directly from Play Store and App Store respectively. However, if you want to have this app on your PC or laptop, the process is a little longer and can potentially expose your computer to malware.
But if you need the Photomath application on your personal computer, just follow the steps outlined down below so you can successfully install the application and use it to get all the help you need with your tough math questions.
- First of all, you will need to find and download an android emulator. An Android emulator is software that simulates Android devices on your computer. With the help of this, you can test and run many different Android applications on your computer.
- The most popular Android emulators are the NoxPlayer and the BlueStacks emulator. You can download either one and use it to run Android applications on your PC or laptop.
- After downloading the Android emulator, and completing the installation, launch the emulator by double-clicking its icon on the desktop.
- Open your Google Play Store through the emulator and click on My Apps.
- In the search bar, type Photomath and press enter or click on the search icon in the bar.
- Download and install the Photomath application on your desktop.
- Double click on the Photomath icon on your screen. Launch it, and get started on your mathematical problems.
Here’s a quick tutorial video for your reference:
Features of Photomath App
Photomath was created by the Deputat family corp in 2014 to provide an easy way to learn and solve mathematics for both teachers and students. Some key features that helped in making this application a raging success include the following:
No internet usage: When using the Photomath application, the problems that are equation-based can be solved even without being connected to the internet. This is because the application has a built-in method to recognize and solve specific questions that use the same equation. By substituting the numbers, the application solves your problem even in the absence of internet access.
Free to use: The Photomath application is absolutely free to use. Having this app is like having a personal tutor who can help you out with your math homework that too free of charge! Not only that, this app can be used whenever you want without having restrictions that come with hiring a tutor or joining an academy.
Multiple explanations: Photomath provides you with multiple step-by-step explanations of the solution, and more than one solving method so you can choose one that is the easiest for you to understand. Additionally, Photomath comes equipped with a Multi-functional scientific calculator that can further enhance your experience of learning.
Interactive learning: One of the latest and very popular features of the Photomath application is the exclusive How-To animations that depict in a simple and easy-to-understand manner how a question was solved. Not only that, the application is loaded with simple yet high-yield graphs that will greatly improve your learning process.
Photomath Alternatives that you can Install
Some other applications offer similar math solving features. These include:
MyScript Calculator app supports basic arithmetic, square roots, and cube roots but does not solve trigonometry, logarithmic equations, and percentages. This application also recognizes handwritten questions. MyScript Calculator turns your device into an interactive piece of paper. All you need to do is write an equation and it gives you the result within seconds.
Not only that, but you can also reuse previous results with a drag and drop feature. This app is based on MyScript Interactive Ink, which is the next step for digital ink and has received much praise worldwide.
iMathematics is available for free on Apple and Android. All you need to do is type in an equation and the app will solve it for you. Additional features that support more complex problems can be unlocked with an in-app purchase. iMathematics allows you to learn quickly from over 70 topics, with clear and simple explanations and easy formulas.
It also includes theorems, definitions, and properties as well as practice with various interactive calculators, like the advanced calculator, the Fraction Approximator, and the Equation Solver.
Mathway offers a useful math calculator for algebra, graphing, and calculus. Simply point your camera and take a picture of your math homework question for step-by-step answers. From basic algebra to calculus, arithmetic, geometry, and trigonometry, Mathway instantly solves your most difficult math problems.
This app has millions of users and even more; solved math problems on its interface. It’s like having a private tutor in your hand, providing instant homework help whenever you need it.
Frequently Asked Questions about Photomath
Q1. Is Photomath free?
Ans: Yes, Photomath is free to download and use on both Android and Apple devices.
Q2. Does Photomath need WiFi?
Ans: Several features can be used without the internet. However, some features also need the internet to run.
Q3. What can Photomath solve?
Ans: Photomath currently supports basic mathematics/arithmetics, Algebra, Geometry, Trigonometry, Calculus, and Statistics.
Q4. Can Photomath solve word problems?
Ans: If you purchase Photomath plus, the additional features can solve a wide range of word problems.


Bethe ansatz for Yangian invariants
Humboldt-Universität zu Berlin
Vendredi 07/06/2013, 14:15-15:15
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
Inspired by Baxter's perimeter Bethe ansatz I present a method to construct Yangian invariants. The condition for Yangian invariance is formulated as an eigenvalue problem that can be solved using Bethe ansatz techniques. The rather general principle is being worked out for rational inhomogeneous spin-chains with finite-dimensional representations of gl(n) in the quantum space. Using the algebraic Bethe ansatz I derive Bethe ansatz equations for Yangian invariants and study the corresponding on-shell Bethe vectors. They can be interpreted as partition functions of certain vertex models.


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Overview: What Is a Conic Section? Conic sections are the curves formed when a right circular cone intersects with a plane. The angle that the plane makes when it “cuts through” the cone determines the shape of the section.
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Over 12 games, a baseball team scored an average of 6 runs per game. If their average number of runs in the first 10 games was 5 and they scored the same number of runs in each of the last two games, how many runs did they score in the last game? Please explain how you arrived at your answer. Step-by-step procedures will be greatly appreciated.
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Public radio network WAMC has a program devoted to women in science, technology, engineering and mathematics aptly called Women in Science ON THE AIR! They've put the show online, so you don't even need to be in the network's listener area to hear the program. From their site description:
The Women In Science, Technology, Engineering and Mathematics ON THE AIR! website is an audio resource for young girls, young women, parents, middle and high school teachers, college professors, guidance counselors, researchers, organizational leaders, and anyone interested in learning more about the past, present and future role of women in science and technology education, fields and careers.There are several different (but overlapping) program series they have available:
- Sounds of Progress: "a series of radio stories that highlight the changing role of girls and women in science, technology, engineering and mathematics (STEM)."
- Powerful Signals: "a special radio series highlighting the role of women and girls in science, technology, engineering, and mathematics (STEM)."
- Her-Story Then and Now: profiles women scientists and engineers, past and present
- The Tech Club: "Aspiring teenage scientists, Emily Lescak and Ivy Hughes join Mary Darcy, Producer and Host of 51% and talk with 54 women in science and technology about why and how each of these women chose her career, what she does each day and what she hopes to accomplish in her work."
Tags: women in science, radio, WAMC


...Knowledge is power! My approach to working one on one with a student is fairly literal and my style is to demand that a student focuses on the task of learning at hand. I have a broad educational background in math, writing, science and music.
Breckenridge, COView Tutor
...After this I started another private tutoring firm and tutored grade school and high school students in Georgetown in Math, Latin, English, and various Sciences. I have worked with students of all ages and backgrounds so I feel confident in being able to teach you or your child in a personally t...
Avon, COView Tutor
...By giving the topics purpose, you can get a better understanding of why we do what we do. Hopefully, this approach will help students in their study of precalculus. I am a retired Algebra 2 and Pre-Calculus teacher.
Evergreen, COView Tutor


Modern scientific and engineering computations are based on a numerical study of applied mathematical models. Mathematical models involve linear or nonlinear equations as well as systems of ordinary differential equations (ODEs). But in the majority of cases, mathematical models consist of systems of partial differential equations (PDEs), which may be time-dependent as well as nonlinear and, moreover, they may be strongly coupled with each other. In addition, these equations are supplemented with the appropriate boundary and initial conditions. To obtain high-fidelity numerical results of practical interest, it is necessary to find solutions of boundary value problems in complicated computational domains.
The existing literature that discusses problems of scientific and engineering computations, in fact, does not reflect the up-to-date realities. Practically, we have books on numerical methods somehow adapted to the needs of the reader. They focus on scientific and engineering computations from the viewpoint of specialists in numerical methods and computational mathematics. This approach includes developing a numerical method, programming a code, performing computations, and processing numerical results; all of these steps are carried out by the readers themselves. This methodology is appropriate for solving rather simple problems and suggests the sufficient universality of the reader, which is quite rare.
Applied software must reflect the state-of-the-art in numerical methods, programming techniques, and the efficient use of computing systems. This can be achieved with a component-based software development. In this approach, after a modular analysis, a mathematical model is divided into basic computational subproblems, and then an algorithmic interface is organized between them. The solution of the subproblems is implemented using standard computational components of scientific and engineering software. Software packages and modules for pre- and post-processing of problem data may also be treated as components of the developed applied software. The above-mentioned problem-oriented computational components are oriented towards solving typical problems of computational mathematics, and they are developed by experts in numerical methods and programming techniques. The last condition ensures the quality of the developed product when we employ modern computing systems.
Applied software is developed using certain standards and agreements. It concerns, in particular, a programming language. For a long time the software for scientific and engineering computations was implemented in the programming language Fortran. The main advantage of Fortran is a large number of programs and libraries written in it, which are often freely available with a source code and documentation. Nowadays, the situation is changing in favour of other programming languages, especially in favour of C and C++. At present, new mathematical libraries and particular components are usually written in C/C++. Moreover, many well-proven applied software projects developed early in Fortran were rewritten in C/C++.
In research projects, we traditionally focus on using free and open source software (FOSS). It is especially suitable for educational process. From our point of view, the natural business model should be based on payment to an educational institution for training potential users of proprietary software, but in reality, it seems to be quite the contrary. The second requirement, in our mind, is portability, i.e. software should work on various hardware platforms and/or operating systems. In more exact terms, programming languages, available libraries, and applied software should be cross-platform.
Another important issue is related to multiprocessor computers. Applied software for multiprocessor computing systems with shared memory (multicore computers) is developed using OpenMP. For systems with distributed memory (clusters), the standard programming technique is MPI. Applied problems that are governed by PDEs can be solved on parallel computers using the library PETSc.
These key thoughts have determined the structure of this book and its general direction of designing modern applied software. We describe the basic elements of present computational technologies that use the algorithmic languages C/C++. The emphasis is on GNU compilers and libraries as well as FOSS for the solution of computational mathematics problems and visualization of the obtained data. This set of development tools in other circumstances might be slightly different, but this does not change the general orientation.
All the questions of the numerical solution of applied problems on parallel computing systems are discussed in the second volume.
The book was prepared by a team of researchers from the Center of Computational Technologies, M. K. Ammosov North-Eastern Federal University, Yakutsk, Russia, and scientists from the Nuclear Safety Institute, Russian Academy of Sciences, Moscow, Russia. We hope that this book will be useful for students and specialists who solve their engineering and scientific problems using numerical methods. We would be grateful for any any constructive comments on the book.
Petr N. Vabishchevich
Moscow and Yakutsk, March 2014.